-m- 


UNIVERSITY  .OF    CALIFORNIA. 

Receded  ^fytu£  _',  i 

Accessions  No.  ^^>^^/ '     Shelf  No. 


AZIMUTH. 


A  TREATISE  ON  THIS  SUBJECT, 


WITH  A  STUDY  OF  THE  ASTRONOMICAL  TRIANGLE,  AND  OF  THE 
EFFECT  OF  ERRORS  IN  THE  DATA. 


BY 


JOSEPH    EDGAR   CRAIG, 

LIEUTENANT-COMMANDER,  U.  S.  NAVY. 


NEW  YORK: 

JOHN    WILEY  &  SONS, 

15  ASTOR  PLACE. 

1887. 


COPYRIGHT,   1887, 
BY  JOSEPH    EDGAR  CRAIG. 


CONTENTS. 


PART  I. 

FAGBS 

INTRODUCTION     -------  .       -       -       -  ._.-         i_6 

PART  II. 

DEDUCTION  OF   FORMULAS.  FOR  THE  DETERMINATION  OF  AZIMUTH. 

Altitude-azimuth  7-8 

Time-azimuth        ---__-____----         -_.  8-9 

Time-altitude-azimuth          ---____-_.____-  9-10 

Horizon-azimuth  -----------------  10 

Time-altitude-latitude-azimuth     -------------  10 

PART  III. 

DIFFERENTIAL   VARIATIONS   IN   THE  ASTRONOMICAL  TRIANGLE  WITH  REFERENCE  TO   AZIMUTH. 

General  remarks  upon  errors  and  changes  -  _-.-__-.--       11-12 

Deduction  of  differential  equations  in  the  problems  of 

Altitude-azimuth   ----------------      12-13 

Time-azimuth         --..--_-___--_-_       14-15 
Time-altitude-azimuth  -        -        -        -        -        -        -        -        -        -        -        -        -        -        -15-16 

Horizon-azimuth  ----------------      16-18 

PART  IV. 

CONSIDERATIONS    AFFECTING    THE    EQUATIONS    OF    MAXIMUM    AND    MINIMUM    ERRORS,   AND   RESPECTING 

THE  CURVES  OF  THESE   ERRORS. 

Method  of  reckoning  the  angles  /,  Z,  and  q  __.___.      19-20 

Remarks  on  the  terms  of  the  equations,  co-ordinates,  and  diagrams  -------      20-21 

Explanation  of  the  terms  maximum  and  minimum  as  used  in  this  treatise  21 

dZ  dZ  dTj 

Tabular  statement  of  equivalent  expressions  for  the  value  of  -JT,  — r— ,  -rr,  etc.,   in    the    several    azi- 
muth problems    ----------------  22 

Other  useful  equations          --.-___-__-----  23 

PART  V. 

DETERMINATION,  BY  INSPECTION,  OF  THE   MOST    FAVORABLE  AND   THE   LEAST  FAVORABLE   POSITIONS  OF  A 
GIVEN    BODY    FOR   OBSERVATION   IN   A   GIVEN   LATITUDE. 

Altitude-azimuth  -  __.__-___--  25-26 

Time-azimuth        ____.__--____----  26-27 

Time  altitude-azimuth  -         -         _____-____-_--  27-28 
Remarks  on  the  algebraical  determination  of  the  expressions  of  maximum  and  minimum  errors  in  the 

computed  azimuth        -__-___--  _____  28-29 

Remarks  on  tracing  the  curves  and  a  summary  of  forms  of  equation  to  the  loci  30 


iv  CONTENTS. 

PART  VI. 

EQUATIONS    TO    THE    LOCI    OF  MAXIMUM    AND  MINIMUM    ERRORS    IN  THE  COMPUTED  AZIMUTH  DUE  TO 

ERRORS  IN  THE  DATA. 

PAGES 

Locus  No.  i.  Altitude-azimuth  error  in  h    •  •-'.-'.*,_* •        -  31-34 

Formulas  of  transformation  to  obtain  the  equations  of  the  projected  curves       -        -        -        -        -  34-36 

Locus  No.  2.  Altitude-azimuth  error  in/,------------  36-37 

Locus  No.  3.  Altitude-azimuth  error  in  </------        ...--._  37 

Locus  No.  4.  Time-azimuth  error  in/           -        -        -        -        -        -        -        -        -    •    -  "     -        -  37-41 

Locus  No.  5.  Time  azimuth  error  in  Z.          ._.._-----.-  41-42 

Locus  No.  6.  Time-azimuth  error  in  d  -        -        -        -        -        -        -        -        -        -        -        -        -  42~44 

Locus  No.  7.  Time-altitude-azimuth  error  in  h     -        -        -        -        -        -        -        -        -        -        -  44-46 

Locus  No.  8.  Time-altitude-azimuth  error  in/-----------  46-48 

Locus  No.  9.  Time-altitude-azimuth  error  in  d    -        -        -        -        -        -        -        -        -        -        -  48-49 

Locus  No.  10.  Time-azimuth  and  altitude-azimuth  for  error  in  L  giving  equal  numerical  values  to  the 

error  in  azimuth 49~5o 

First.  Identical  errors,  signs  alike                                                                                         -        -  50-51 

Second.  Equal  errors,  numerical ;  opposite  signs  51-53 

PART  VII. 

ANALYSIS  OF  THE  EQUATIONS  TO  THE   LOCI   AND  THE  TRACING  OF  THE  CURVES. 

Introductory  remarks     -        -  54~55 

Locus  No.  i.  Altitude-azimuth  error  in  ^ - 55-63 

Locus  No.  2.  Altitude-azimuth  error  in  Z    - -        -        -        -        -  63-64 

Locus  No.  3.  Altitude-azimuth  error  in  d------------  64 

Locus  No.  4.  Time-azimuth  error  in/-        -        -        -        -        -        -        -        -        -        -        -        -  64-70 

Locus  No.  5.  Time-azimuth  error  in  Z  -        - 70-73 

Locus  No.  6.  Time-azimuth  error  in  d - 73~76 

Locus  No.  7.  Time-altitude-azimuth  error  in  £----•---•--  76-80 

Locus  No.  8.  Time-altitude-azimuth  error  in/ 80-81 

Locus  No.  9.  Time-altitude-azimuth  error  in  d 81 

Locus  No.  10.  Time-azimuth  and  altitude-azimuth  for  error  in  L  giving  the  same  numerical  error  in 

the  computed  azimuth                                                                                --....  81-83 

Concluding  remarks -_-....  83-84 

APPENDIX. 

Remarks  embracing  a  scheme  for  deriving  a  great  variety  of  loci  from  the  variations  of  the  spherical 

triangle .  85^3 

Loci  in  time-altitude-latitude-azimuth -....  93-97 

Loci  in  time-sight          --.._ 97-101 

Notice  of  error  and  its  correction          ---.._ 101-102 

Loci  in  the  problem  of  latitude  by  altitude  at  any  time        --.._..._  103-104 

Loci  in  the  problem  of  computing  the  altitude --___.  104-105 

Explanation  of  the  plates      ---------____._  IO6 


'*m  ; 


AZIMUTH. 


PART  I. 
INTRODUCTION. 


1.  IT  is  purposed  in  the  following  notes  to  suggest  a  study  of  the  astronomical  triangle 
with  respect  to  the  azimuth  problem,  supplementing  the  teaching  of  the  text-books ;  also,  to 
point  out  false  teaching  of  the  latter  on  some  points  referring  to  the  most  favorable  condi- 
tions of  observation  to  reduce  to  a  minimum  in  the  computed  azimuth  the  effect  of  small 
errors  in  the  data. 

A  thorough  study  of  the  astronomical  triangle,  viewing  it  in  all  possible  aspects,  in  the 
several  problems  of  nautical  astronomy — time,  latitude,  and  azimuth — is  instructive  as  an 
exercise,  even  when  investigation  is  extended  beyond  the  restricting  limits  imposed  by  the 
practical  problem ;  that  is,  if  free  to  select  any  part  of  the  triangle,  to  take  successively  each 
part  as  fixed,  and  then  to  seek  the  best  condition  for  all  the  remaining  parts  ;  one  of  these 
predominating  in  influencing  a  choice,  and  so  making  a  third  part  subservient  to  fixing  the 
others. 

2.  The  problem  presented  in  practice  restricts  the  observer  to  a  particular  spot,  and 
many  times  the  observer  limits  himself  to  the  observation  of  a  particular  celestial  body :  then 
the  latitude  and  the  declination  are  fixed,  and  it  remains  to  find  the  value  of  some  other 
one  part  affording  the  most   favorable  condition   for  the    observation    of   the  body  when 
considering  the  error  in  each  datum  taken  separately ;  and,  finally,  by  exercise  of  the  judg- 
ment, to  choose  the  condition  giving  the  best  result  when  all  the  errrors.in  the  data  are 
considered.      In    practice,  to  know   when    to   observe,  the  most    convenient   third   part   to 
find    is  either   the  hour-angle  or    the   altitude  ;   primarily  the    former,   but,   for  discussion, 
sometimes  the  latter ;  from  which  is  easily  found  the  hour-angle  to  give  the  time  for  the 
observation. 

If  not  restricted  to  the  employment  of  a  particular  body,  the  first  step  will  be  to  select 
the  best  one  from  several  bodies  that  are  favorably  situated  for  observation. 

3.  Since  the  azimuth  of  a  heavenly  body  is  determined  for  the  purpose  of  ascertaining 
(i)  at  sea  the  compass  error,  thence  the  variation  of  the  magnetic   needle  or  the  deviation 
caused  by  the  iron  in  the  ship,  or  both  variation  and  deviation ;  (2)  on  board  ship,  the  true 
bearing  of  some  point  visible  on  land  ;  or  (3)  on  land  the  same,  thence  the  direction  of  the 
meridian — the  best  practical  conditions  should  be  sought. 

The  lower  the  altitude  the  better  for  an  observation  of  the  compass  azimuth  ;  an 
altitude  of  the  heavenly  body  equal  to  that  of  the  object  on  land  being  the  best  for  the 
determination  of  the  horizontal  angle  between  the  two  objects  by  means  of  the  theodolite. 


2  AZIMUTH. 

or  for  the  observation  of  an  astronomical  bearing  (sextant  being  used)  from  which  the 
horizontal  angle  is  to  be  computed :  because  then  the  error  of  level  will  cause  the  least 
effect  in  observations  with  the  compass  and  the  theodolite,  and  the  error  in  the  observed 
sextant-angle  will  be  multiplied  less  the  more  nearly  the  altitudes  agree. 

Since  the  terrestrial  object  should  be  near  the  horizon,  we  may  say,  in  general,  the  lower 
the  altitude  the  better. 

On  the  other  hand,  if  altitude  enters  the  data,  the  lower  the  altitude  the  more  uncertain 
the  refraction,  and  the  computed  azimuth  will  be  correspondingly  affected  by  error  in  alti- 
tude. In  this  case,  then,  very  low  altitudes,  and  in  all  cases  very  high  altitudes,  should  be 
avoided,  whatever  conditions  are  given  from  the  astronomical  triangle  as  theoretically  the 
most  favorable  for  precision  in  the  computed  azimuth.  Intelligent  discrimination  should  be 
exercised  by  the  observer  to  effect  a  compromise  between  the  theoretical  and  the  practical 
advantages. 

4.  In  ordinary  circumstances  at  sea  it  is  unnecessary  to  exact  the  best  condition  when 
great  labor  would  attend  ascertaining  it ;  as,  for  instance,  in  the  case  of  a  body  that  crosses 
the  prime  vertical  above  the  horizon,  to  find  the  best  position  in  which  to  observe  it,  to  make 
the  effect  of  a  small  error  in  altitude  the  least ;  for  to  do  this  requires  the  solution  of  a  trig- 
onometric equation  of  the  fourth  degree.   Nevertheless,  very  bad  conditions  should  be  avoided. 

In  the  case  of  serial  time-azimuths  to  determine  the  deviation  of  the  magnetic  needle, 
considerable  work  may  properly  be  demanded  to  ensure  taking  observations  extending  over 
the  most  favorable  time,  on  each  side ;  for,  granting  the  impossibility  of  obtaining  very  nice 
results  from  observations  made  at  sea,  yet  as  good  work  as  is  practicable  should  be  done, 
reducing  the  errors  as  much  as  possible. 

In  a  survey  on  land,  with  so  much  better  conditions,  errors  may  be  eliminated  to  a  great 
degree ;  and  the  nearest  approach  to  accuracy  that  can  be  made  should  be  insisted  on,  even 
though  much  labor  attends  the  finding  of  the  best  conditions  for  observation. 

5.  The  truth  should  be  known,  notwithstanding  the  knowledge  of  it  may  not  always  be 
put  to  effective  use ;  therefore  erroneous  assertions  in  well-known  and  much-used  text-books 
should  not  be  perpetuated. 

In  the  following  extracts  the  italics  are  the  writer's,  not  the  authors' : 
MAYNE'S  MARINE  SURVEYING,  page  90,  treating  of  true  bearings,  says:  "  Firstly,  the 
body  should  be  rising  or  falling  rapidly,  when  its  movement  in  azimuth  will  also  be  rapid."  * 

In  the  first  place,  slow  should  be  substituted  for  rapid;  but,  even  with  this  correction 
made,  the  words  in  italics  are  not  true  in  reference  to  any  body  not  on  the  equator  f  that,  in 
its  diurnal  course,  crosses  the  prime  vertical.  Though  the  author  favors  the  method  of  time- 
azimuth,  he  makes  his  remarks  apply  to  the  altitude-azimuth  as  well.  Now,  it  is  not  the 

*  "  In  other  words,  the  nearer  the  prime  vertical  the  better." 

f  If  the  declination  of  the  body  is  zero,  the  most  favorable  position,  theoretically  speaking,  is  when  on  the 
prime  vertical — that  is,  at  the  intersection  of  the  horizon,  equator,  and  prime  vertical  (t  =  6h,  h  =  o,  Z  =  90°). 

This  one  exception  must  be  understood  throughout,  in  the  statements  denying  that  the  position  of  the  body  when 
crossing  the  prime  vertical  is  the  most  favorable  and  asserting  that  on  the  six-hour  angle  is  a  better  position.  In  this 
case  /  =  90°  and  Z—  90°  coexist  and  give  the  best  position.  But,  if  understood,  It  will  not  be  necessary  to  repeat 
that  this  single  exception  exists. 

When  d=  L,  the  body  not  crossing  is  in  the  best  position,  theoretically,  when  on  the  prime  vertical. 


AZIMUTH. 

ratio  existing  between  change  of  altitude  and  change  of  time  that  should  be  considered; 
but,  in  the  case  of  time-azimuth,  the  ratio  of  change  of  time  to  change  of  bearing,  and,  in 
altitude-azimuth,  the  ratio  of  change  of  altitude  to  change  of  bearing.  With  either  method 
employed,  so  far  as  error  in  time  or  error  in  altitude  is  concerned,  the  body  will  be  more 
favorably  situated  at  the  point  of  crossing  the  six-hour  circle  than  at  transit  on  the  prime 
vertical  (see  articles  43  and  45).  But  the  best  position  lies  between  these  two  points,  in  the 
case  of  altitude-azimuth  for  all  latitudes  ;  and  in  time-azimuth  for  all  latitudes  less  than  45°, 
while  with  increasing  latitudes  the  best  position  for  certain  declinations  approaches  the 
meridian  in  bearing,  and  finally  reaches  it  (see  art.  95). 

While  considerable  labor  may  be  required  to  determine  this  position  in  order  to  seize  the 
observation  there,  it  can  be  taken  at  the  first  point  mentioned  (t  =  90)  with  less  trouble  than 
if  observed  at  the  second  (Z  =  90),  often  with  the  further  advantage  of  having  a  better  alti- 
tude for  observing  the  compass-azimuth  and  the  astronomical  bearing  of  the  object  on  the 
earth.  This  is  worth  knowing  in  cases  where  the  labor  of  finding  the  best  position  may 
reasonably  be  dispensed  with. 

6.  CHAUVENET'S  ASTRONOMY,  vol.  L,  page  431,  treating  of  true  bearings,  employing 
the  method  of  altitude-azimuth,  says:  "From  the  first  equation  of  (50),  0  and  $  being  con- 

dk 

stant,  dA  — j— ,  and  therefore  an  error  in  the  observed  altitude  will  have  the  least 

cos  h  tan  q 

effect  upon  the  computed  azimuth  when  tan  q  is  a  maximum ;  that  is,  when  the  star  is  on  the 
Prime  vertical.  Therefore,  in  the  practice  of  the  preceding  method  the  star  should  be  as  far 
from  the  meridian  as  possible"  [far,  in  bearing  or  azimuth]. 

For  a  star  that  crosses  the  prime  vertical  the  quotation  in  italics  is  not  true.  Instead  of 
"  when  tan  q  is  a  maximum"  read  when  tan  q  cos  h  is  a  maximum,  and  reject  all  that  follows  ; 
and  this  without  disputing  the  correctness  of  the  language  intended  to  convey  the  meaning. 

7.  But  it  may  be  pertinent  to  remark  here  that  text-writers  have  a  careless  habit  of  using 
the  words  far  from  and  near  to,  referring  to  the  meridian  and  the  prime  vertical.     They 
know  what  Jthey  mean  to  say,  but  may  mislead  the  novice.     The  primary  idea  conveyed  by 
this  language  is  distance,  absolute,  which  in  this  particular  case  would  place  the  body  on  the 
six-hour  circle,  while  in  the  p.  v.  is   meant;  the   secondary  idea,  that  of   time-measurement 
from  the  upper  branch   of  the  meridian,  which  would  place  the  body  in  the  horizon ;  and, 
last  of  all,  the  idea  of  angular  distance  in  azimuth  (or  bearing),  yet  this  is  what  is  meant. 
Say  far  from  (or  near)  in  azimuth,  or  in  bearing ;  or  else  say  when  the  body  bears  most  nearly 
east  and  west,  or  north  and  south,  as  the  case  may  be.      As  an  instance  of  carelessness,  a 
standard  work — and  many  others  err  in  the  same  way — states  that  for  an  error  in  altitude  in 
the  time-sight  the  effect  on  the  computed  time  will  be  least  when  the  body  is  observed 
nearest  the  prime  vertical.     Now,  a  body  whose  declination   is  greater  than  the  latitude  of 
the  same  name  will  be  nearest  the  prime  vertical  when  on   the  meridian,  at  upper  culmina- 
tion, when  the  zenith  distance  is  least ;  and  this  is  the  worst  possible  position  when  consider- 
ing errors  in  altitude  not  only,  but  in  latitude  and  declination  as  well.     Growing  out  of  this 
carelessness  on  the  part  of  intelligent  authors,  many  text-books  on  navigation,  compiled  by 
writers  without  thinking,  publish  the  absurd  statement  that  the   method  of  single-altitude 
observation  for  latitude  should  be  used  only  when  the  heavenly  body  is  within  an  hour  (or 
thereabouts)  of  the  meridian.     Believing  this,  what  would  become  of  the  reputation  of  the 


4  AZIMUTH. 

pole-star,  earned  by  its  unvarying  good  character  foi   a  latitude-observation  throughout  its 
diurnal  course? 

8.  COFFIN'S  NAVIGATION  AND  NAUTICAL  ASTRONOMY,  Fifth  Edition,  page  277,  treat- 
ing of  true  bearings,  says:  "  Zs  =  NZM,  the  azimuth  of  the  celestial  body,  maybe  found 
from  an  observed  altitude  (Prob.  40),  or  from  the  local  time  (Prob.  38). 

"  In  the  first  case  the  most  favorable  position  is  on  or  nearest  the  prime  vertical ;  for  then 
the  azimuth  changes  most  slowly  with  the  altitude.  In  the  latter,  positions  near  the  meridian 
may  also  be  successfully  used." 

For  a  body  that  crosses  the  prime  vertical  the  italicized  statement  is  erroneous.  While 
the  last  part  of  the  quotation  (not  in  italics),  in  respect  to  error  in  time,  is  true ;  yet  the  word 
also  implies  that  the  best  position  is  on  the  prime  vertical,  which  is  not  true.  It  is  worth 
remarking  in  this  case  (time-azimuth)  that  an  error  in  latitude  will  produce  a  maximum  effect 
in  the  computed  azimuth  if  the  body  is  observed  at  a  particular  point  lying  between  the 
intersections  of  the  prime  vertical  and  the  meridian  with  the  diurnal  circle ;  and  this  point 
should  be  avoided.  Dependent  on  the  relative  values  of  the  latitude  and  declination,  it  may 
be,  in  bearing,  far  from  or  near  to  the  meridian. 

9.  MARINE  SURVEYING,  U.  S.  NAVAL  ACADEMY,  page  25,  on  the  subject  of  true  bear- 
ings, treating  of  time-azimuth,  to  be  superseded  by  altitude-azimuth  if  the  time  is  not   accu- 
rately known  (page  26),  says:  "  The  observation  should  be   made  when  the  heavenly  body  is 
near  the  prime  vertical,  provided  it  has  not  too  great  an  altitude  at  that  time." 

10.  Even  if  on  the  prime  vertical  were  the  best  position  considering  error  in  altitude  alone 
or  time  alone,  there  is  another  datum — the  latitude — that  may  have   a  considerable   error, 
while  both  the  time  and  altitude  may  be  determined  to  a  comparatively  great  degree  of  ac- 
curacy.    In  this   case  if  the  method  of  altitude-azimuth  is  chosen,  a  small  error  in  latitude 
will  produce  no  error  in  the  computed  azimuth,  provided  the   body  is  observed  when  on  the 
six-hour  circle  ;  hence,  this  is  the  most  favorable  position  for  error  in  latitude,  and  it  is  a 
better  position  than  that  on  the  prime  vertical  for  error  in  altitude. 

If  time-azimuth  is  used,  a  rising-and-setting  body  will  be  in  the  best  position,  respecting 
error  in  latitude,  when  in  the  horizon — the  effect  of  this  error  then  reducing  to  zero.  And 
this  position  may  be  better  than  that  on  the  prime  vertical  for  error  in  time — depending  on 
the  relative  values  of  the  latitude  and  declination. 

11.  Not  unfrequently  in  surveying,  a  true  bearing  may  be  demanded   for  preliminary 
plotting  before  either  time  or  latitude  has  been  accurately  determined.     The  latter,  as  well 
as  the  longitude,  will  in  many  cases  be  known  approximately:  an  error  in  the  assumed  latitude 
then    need    not  deter   the  surveyor  from  using  the  altitude-azimuth,  with  excellent  results, 
when  the  body  is  on  or  near  the  six-hour  circle. 

Circumstances  may  have  foiled  the  observer's  attempts  to  determine  the  latitude,  and 
yet  have  permitted  excellent  observations  of  equal-altitudes  for  time.  He  then  can  choose 
between  the  two  methods  for  azimuth ;  or  a  third  method,  the  time-altitude-azimuth,  in  which 
latitude  does  not  enter  as  a  part,  may  be  employed,  provided  the  altitude  is  not  very  great 
when  the  body  is  observed  near  the  meridian  in  bearing,  and  the  hour-angle  is  not  very  small. 
This  method  may  often  be  employed  with  excellent  results  in  observations  of  the  sun  when 
the  declination  and  the  latitude  have  contrary  names  (see  art.  14),  and  still  better  when  a 
close  circumpolar  star  is  observed. 


AZIMUTH.  5 

12.  If,  at  a  particular  instant,  an   observation  for  azimuth   is  wanted,  and  the  time  is 
accurately  known,  the  first  thought  will  be  to  use  the  convenient  time-azimuth  tables;  or,  if 
the  time  is  not  well  known,  to  employ  the  method  of  altitude-azimuth.     But  supposing  both 
of  these  data  accurately  known  and  the  latitude  uncertain,  which  of  the  two  methods  shall 
be  chosen  ? 

Even  admitting  comparatively  slight  errors  in  altitude  and  time,  which  shall  be  the 
choice  ? 

If  the  last-mentioned  errors  should  have  exactly  the  same  value  (measured  by  the  same 
unit,  in  arc),  and  considering  any  error  existing  in  the  declination,  but  that  no  error  in  the 
latitude  exists,  the  method  of  time-azimuth  would  be  the  better  for  the  observation  taken  at 
any  time  whatever, — excepting  at  the  single  instant  when  the  parallactic  angle  might  have 
the  value  of  ninety  degrees  (q  =  90°),  possible  only  when  the  declination  exceeds  the  lati- 
tude ;  for  then  the  two  methods  would  give  identical  values  to  the  total  error  in  the  com- 
puted azimuth.  Therefore,  if  the  latitude  could  always  be  known  exactly,  and  time  and  alti- 
tude were  equally  trustworthy,  the  choice  should  always  fall  on  the  time-azimuth. 

13.  Admitting  the  uncertain  and  considerable  error  in  latitude,  while  the  conditions  of 
the  preceding  paragraph  remain  otherwise  unchanged,  the  effect  of  this  error  alone  must  be 
the  criterion  for  choosing  the  method  to  use. 

For  a  circumpolar  star  whose  declination  is  greater  than  the  latitude,  the  time-azimuth 
will  be  the  better  from  the  time  of  lower  culmination  until  a  point  e  is  reached  by  the  star 
before  it  arrives  on  the  six-hour  circle  (/  =  90°) ;  the  altitude-azimuth  will  be  better  from  this 
point  e  until  the  star  attains  its  greatest  elongation  (q  =  90°);  thence,  until  its  upper  culmina- 
tion, the  time-azimuth  the  better.*  If  not  a  circumpolar  star,  the  time-azimuth  is  better  from 
the  time  of  rising  until  at  e'  before  reaching  t  =  90° ;  thenceforward  to  the  meridian  the  same 
as  for  a  circumpolar  star. 

If  the  declination  is  less  than  the  latitude  of  the  same  name,  by  substituting  "  reaches 
the  prime  vertical  (.2"  =90°)"  for  "attains  its  greatest  elongation  (^  :=  90°),"  the  foregoing 
statement  will  apply. 

If  the  declination  is  zero,  or  has  the  contrary  name  to  that  of  the  latitude,  the  time- 
azimuth  will  be  the  more  favorable  during  the  whole  time  of  visibility  of  the  star. 

14.  The    method    of  time-altitude-azimuth,    though   generally  ignored  by  navigators,    is 
deserving  of  attention,  as  giving  advantage  in  some  circumstances   over  both  the  time-azi- 
muth, and  the  altitude-azimuth.     It  should  not  be  used  when  Z  is  near  90°,  /  near  o°,  or  with 
a  high  altitude  (see  art.  11)  ;  but  at  sea,  when  the  ship  and  the  sun  are  on   opposite  sides  of 
the  equator,  there  is  a  wide  extent  of  cruising-space  where,  at  the  best,  poor  conditions  are 
given  for  observations  of  the  sun   for  time.     The  azimuth  determined   by  this  third  method, 
from  the  same  altitude  observed  for  time,  is  more  conveniently  found  than  if  taken  from  the 
time-azimuth  tables,  considering  the  double  interpolation  needed  for  the  actual  latitude  and 
declination.     For,  working  side  by  side  with  the  computation  for  time,  at  the  same  opening  of 
the  logarithmic  tables  required   by  the  latter,  only  two  additional  logarithms  are  needed  to- 
gether with  the  arithmetical  complement  of  one  already  taken  out,  the  sum  of  the  three  giving 

*  Considering  the  eastern  hemisphere  alone,  since  symmetrical  conditions  exist  west  of  the  meridian,  and  limit- 
ing the  angles  to  180°,  as  in  art.  17. 


6  AZIM  UTH. 

the  log  sine  of  the  whole  azimuth.  The  chief  error  will  be  owing  to  the  error  in  the  com- 
puted time,  arising  from  any  existing  error  in  the  latitude ;  but  the  azimuth  taken  from  the 
tables  will  embrace  this  error,  and  the  error  in  the  latitude  itself,  the  latter  not  entering  in 
the  method  under  discussion.  Recourse  may  be  had  to  the  altitude-azimuth,  but  here  the  error 
in  latitude  enters,  and  to  a  much  greater  degree  than  in  the  time-azimuth ;  while  the  work  of 
computation  is  considerably  greater  than  in  the  time-altitude-azimuth.  By  the  latter,  the 
azimuth  being  derived  from  its  sine,  two  values  will  be  given,  greater  and  less,  respectively, 
than  90°;  but  the  conditions  of  observation  readily  determine  the  actual  value. 

15.  The  method  of  horizon  azimuth  or  amplitude,  while  useful  at  sea,  and  very  con- 
venient in  the  case  of  the  sun  on  account  of  the  prepared  tables,  should  not  be  used  in  nice 
work ;  since  the  degree  of  correctness  in  the  result  depends  on  the  approach  to  accuracy  in 
estimating  that  the  body,  whose  image  is  lifted  by  uncertain  refraction,  is  in  the  true  horizon. 
Rejecting  this  fourth  method  in  the  summing-up,  and  noting  that  the  single  case  of  declina- 
tion equal  to  zero  is  excepted  from  the  truth  of  the  statements,  we  have  the  following : 

SUMMARY. 

1st.  As  to  errors  in  declination  and  in  latitude,  the  body  on  the  prime  vertical  is  not  in 
its  most  favorable  position  for  reducing  the  error  in  the  azimuth  computed  by  any  one  of  the 
three  methods. 

2d.  If  a  body  crosses  the  prime  vertical,  the  best  position  is  not  on  it  for  the  error  in  any 
single  datum  in  any  one  of  the  three  methods. 

3d.  For  error  in  /  in  the  time-azimuth  or  for  error  in  h  in  the  altitude-azimuth,  the  body 
that  crosses  the  prime  vertical  is  in  a  less  favorable  position  when  on  the  prime  vertical  than 
when  on  the  six-hour  circle ;  but  the  latter  is  not  the  best  position.  The  best  will  not,  however, 
be  the  same  point  for  error  in  h  that  it  is  for  error  in  /. 


PART  II. 

DEDUCTION    OF    FORMULAS    FOR   THE    DETERMINATION   OF 

AZIMUTH. 


16.  Thus  far  assertions  alone  have  been  given  ;  their  truth  remains  to  be  proved. 
The  notation  used  will  be  as  follows : 

d,  for  the  declination  of  the  body; 

h,  "  "  altitude         "     "       " 

/,  "  "  hour  angle   "     "       " 

Z,  "  "  azimuth        "     "       " 

q,  "  "  parallactic  angle  of  the  body  ; 

/,  or  co-d,  "  "  polar  distance  of  the  body,  =  (90°  —  d),  observing  sign  ; 

z,  or  co-h,  "  "  zenith  distance "     "       "        =  (90°  —  h),         "             " 

L,  "  "  latitude              "     "     place; 

co-L,  "  "  co-latitude         "     "         "      =  (90°  —  L). 

17.  In  computing  the  azimuth  the  hour-angle  t  will  be  (see  art.  34)  regarded  as  positive 
from  o  at  the  upper  culmination  of  the  body  to  180°  at  the  lower  culmination,  whether  to 
the  west  or  east  from  the  meridian;  the  azimuth  Zas  positive  from  o°,  at  the  point  of  the 
horizon  nearest  to  the  elevated  pole  (N.  or  5.  point)  around  to  180°  at  the  opposite  point  of 
the  horizon  (S.  or  JV.  point),  whether  to  the  west  or  to  the  east  ;  the  parallactic  angle  q  as 
always  positive,  o°  up  to  a  possible   180°;  the    latitude   L  as  always  positive  towards  the 
elevated  pole,   from  o°  to  90° ;    the  declination  d  as    positive  if  of   the  same  name  as  the 
latitude,  as  negative  if  of  contrary  name,  o°  to  ±  90°  ;    the  polar  distance  p  as  always  posi- 
tive from  o°  at  the  elevated  pole  to   180°   at  the  depressed  pole;    the  zenith  distance  z  as 
always  positive  from  o°  at  the  zenith  to  180°  at  the  nadir;  the  altitude  h  as  positive  if  above 
the  horizon,  negative  if  below,  o°  to  ±  90°  :  the  co-latitude  co-L  as  always  positive,  o°  to  90°. 

18.  The  following,  well-known  formulas  are  given  for  convenience  of  reference : 

ALTITUDE-AZIMUTH. 
Given  h,  d,  and  Z,  to  find  Z.  1     °>          co-i 

|B/a  Z    ^c°* 

From  the  fundamental  formula  of  trigonometry, 

FIG.  i.  FIG.  2. 

cos  b  =  cos  a  cos  c  -f-  sin  a  sin  c  cos  B,  we  have  ) 
sin  d  =  sin  h  sin  L  -f-  cos  h  cos  L  cos  Z  f 

sin  d  —  sin  h  sin  L 
cos  Z  = -. •= ; (2) 

cos  h  cos  L  ^  ' 


8  AZIMUTH. 

from  which  are  derived 


cos  s  cos  (s  —  p)  1fr  f\ 

when       *  =  i(L  +  *+t):     •    •    •    •    (3) 


/cos  s  sin  (s  —  d ) 

=A/-         -v—  when        j  =  #co-L  +  h-\-  d)  ;      ...     (4) 

y      sin  co-L  cos  A; 


/sin  (s  —  L)  sin  (s  —  K) 

taniZ  =  \/~  —rr^          when         s  =  $(L -4- h -\- p).    ...     (5) 

y        cos  s  cos  (s—p) 

Small  errors  in  the  data  will  have  the  same  effect  on  the  computed  azimuth,  whichever 
formula  is  used.  So  far  as  inexactness  in  the  logarithmic  tables  is  concerned,  formula  (5)  will 
give  the  result  nearest  to  precision,  since  the  tangent  of  an  angle  varies  more  rapidly  than 
either  the  sine  or  the  cosine. 

\iZ~>  90°,  (3)  will  be  better  than  (4);  if  Z  <  90°,  (4)  is  preferable  to  (3);  since  the 
cosine  varies  more  rapidly  than  the  sine  for  an  angle  greater  than  45°,  and  less  rapidly  for  an 
angle  less  than  45°. 

19.  TIME-AZIMUTH. 

Given  /,  L,  and  d,  to  find  Z. 

From  the  fundamental  formula,  we  have 

sin  A  cot  B  =  sin  c  cot  b  —  cos  c  cos  A 
sin  t  cot  Z  =  cos  L  tan  d  —  sin  L  cos  t 


cos  L  tan  d  —  sin  L  cos  t 
cot  Z  = : — ; (7) 

cm   /  v/ 


sn 


from  which  are  derived  tan  0  =  cot  d  cos  /  ; 


cotz=cos(0+£)cgi-'- 

sin  0 
*:„  i,  -  sin(0  +  L)  sin  d 


For  the  solution  of  an  astronomical  bearing — the  altitude  of  the  heavenly  body  not  be- 
ing observed — formula  (10)  will  be  needed  for  finding  the  true  altitude,  from  which  the 
required  apparent  altitude  will  be  determined. 

It  will  be  convenient  to  accept  the  value  of  0  less  than  90°,  taking  care  to  give  it  the 
proper  sign.  There  will  be  no  ambiguity  in  the  value  of  Z  found  from  its  cotangent.  If  the 
latter  is  positive,  the  azimuth  is  less  than  90°  ;  if  negative,  greater  than  90°. 


AZIMUTH.  9 

The  following  formulas,  derived  from  (7),  are  given  in  some  text-books  to  the  exclusion 
of  the  preceding: 


Case  /.—When  5  <  90 


S  =  l>(p  +  co-L)  •    ..........    ,    .  (11) 

D  =  \{p  ~  co-L),  the  greater  minus  the  less;.     .     .  (12) 

tan  X  •=.  sin  D  cosec  5  cot  \t;      .....     ....  (13) 

tan  Y  =  cos  D  sec  5  cot  \t  ;    ..........  (14) 

Z  =  X±  Y.   ...........    .    .     .  (15) 


llp>coL,X+Y=Z  ...........     (16) 

lip<coL,X-Y=Z  ...........     (17) 


Case  //.—When  5  >  90°  : 

(.r-  F)  =  Z.     ...........     (18) 


These  formulas  are  convenient  when  a  series  of  observations  is  to  be  taken,  as  in  the 
case  of  serial  time-azimuths,  therefore  convenient  for  preparing  a  table  of  azimuths  ;  because 
sin  D,  cos  D,  cosec  S,  and  sec  S  have  constant  logarithms  (in  the  case  of  the  sun  regarding  the 
declination  at  the  mean  of  the  times  of  observation  as  constant)  ;  but  for  a  single  observation 
they  give  more  labor  than  (8)  and  (9)  exact,  and  some  trouble  in  freeing  the  result  from 
ambiguity,  whereas  (9)  is  perfectly  clear. 

If,  however,  besides  Z  the  angle  q  were  required  to  be  computed,  formulas  (11)  to  (14) 
would  be  the  most  convenient. 

If  /  >  coL,  for  (12),  write  D  =  %  (p  —  coL)  ; 

for  tan  X,  write  tan  %  (Z  —  q)  }   whence  sum  =  Z; 
for  tan  F,  write  tan  \(Z-\-  q)  f  whence  difference  =  q. 

lip  <  coL, 

for  D,  write  (coL  —  /)  ; 

for  tan  X,  write  tan  £  (q  —  Z},  )  whence  sum  gives  q  ; 

for  tan  F,  write  tan  %(q-\-  Z},  }  and  difference  gives  Z. 

20.  TIME-ALTITUDE-AZIMUTH. 

Given  /,  h,  and  d,  to  find  Z\ 

sin  a  sin  B  =  sin  b  sin  A  | 
cos  h  sin  Z  =  cos  d  sin  /  ) 

sin  t  cos  d  .  , 

sin  Z  =  --  :—  ..............     (20) 

cos  li  ^    ' 


io  AZIMUTH. 

Of  the  two  values  of  Z  given  by  (20),  supplements  of  each  other,  the  proper  one  may  be 
known  from  the  conditions  of  the  case.  If  Z  is  very  near  90°,  and  it  is  doubtful  on  which 
side  of  the  prime  vertical  the  body  lies,  the  ambiguity  may  not  be  removed  ;  but  the  method 
is  inadmissible  then,  since  a  small  error  in  any  given  part  may  make  Z  impossible,  sin  Z 
being  greater  than  unity;  and  in  any  event  such  error  will  produce  in  Z  a.  very  large  error 
(see  art.  14). 

21.  HORIZON-AZIMUTH. 

Given  h  =  o,  d,  and  L,  to  find  Z: 

sin  d 


From  (2),  cosZ:= 


^o7Z 


Attending  to  the  sign  of  d,  no  ambiguity  can  arise. 

22.  A  fifth  method,  which  the  writer  has  never  seen  alluded  to,  may  be  employed.  It 
may  be  termed  TiME-ALTITUDE-LATITUDE-AZIMUTH,  to  distinguish  it  from  that  of  art.  20, 
which,  to  be  precise,  should  be  called  Time-altitude-declination-azimuth. 

Given  t,  h,  and  L,  to  find  Z.  This  method  may  be  resorted  to  in  the  remote  contingency 
of  declination  not  known,  as,  for  instance,  the  mutilation  of  the  ephemeris  so  far  as  concerns 
the  declination  only. 

From  trig.,  sin  B  cot  A  =  sin  c  cot  a  —  cos  c  cos  B\    )  ,     . 

sin  Z  cot  /  =  cos  L  tan  h  —  sin  L  cos  Z;  ) 

whence  cot  $  =  tan  A  cos  c  ;  \ 

cos  $'  =  cos  $  tan  c  cot  a  ;     V    .....     .     ...      (23) 

B  =  &  ±  &  ;  ) 

and 

cos 

Z  = 

Attending  to  the  signs,  there  will  still  be  two  values  found  for  Z.  But,  sine*  negative 
values  of  Z  and  values  greater  than  180°  are  excluded  (art.  \7\  the  proper  value  will  be 
known. 

When  d  is  given  it  will  be  more  trustworthy  than  is  usually  the  Z.,  hence  this  method  is 
not  considered  farther,  though  having  a  place  in  the  problem  of  azimuths. 


cot  L  tan  h  ;         .........      (3, 


PART  III. 

DIFFERENTIAL    VARIATIONS     IN    THE    ASTRONOMICAL    TRI- 
ANGLE  WITH    REFERENCE   TO   AZIMUTH. 

23.  The  error  in  the  computed  azimuth  arising  from  small  errors  in  the  data  may  be  trans- 
lated into  the  change  in  the  value  of  the  azimuth  corresponding  to  small  changes  in  the  data. 

Since  with  three  parts  given  in  the  astronomical  triangle — and  not  less  than  three 
parts — the  remaining  parts  can  be  found,  we  have  in  the  problem  of  azimuths  four  parts  in 
each  of  the  methods  proposed. 

In  order  that  the  astronomical  triangle  shall  change  its  aspect,  not  more  than  two  parts 
can  remain  constant — all  the  others  must  vary.* 

In  the  fundamental  formulas  employed,  differentiating  for  any  two  parts  of  the  triangle 
as  variable,  the  other  two  being  constant,  the  change  in  one  variable  may  be  found  in  some 
terms  of  the  other  variable  and  the  two  constants,  and  in  a  simpler  form,  sometimes,  by 
employing  parts  of  the  triangle  that  do  not  enter  the  problem.  For  the  investigation  of 
maximum  and  minimum  effects  of  errors,  these  equivalent  expressions  may  be  used  with 
advantage. 

24.  In  this  problem,  since  Z  is  regarded  always  as  one  of  the  variables,  we  shall  find  the 
expression  for  the  approximate  error  in  Z  corresponding  to  a  small  error  in  each  of  the  given 
parts,  taken  separately,  the  other  two  parts  being  regarded  as  constant  for  that  occasion. 

The  expression  for  the  total  error  in  Z  will  be,  for  practical  purposes,  the  algebraic  sum 
of  the  three  errors  thus  found  ;  that  is,  the  total  differential  equals  the  sum  of  the  partial 
derivatives. 

25.  In  each  case  the  expression  for  error  depends  on  the  particular  parts  of  the  triangle 
used,  both  the  variables  and  the  two  constants ;  the  remaining  two  parts,  though  they  do 
not  enter  the  formula,  vary  with  the  variables  employed. 

For  illustration,  the  error  in  the  computed  azimuth  owing  to  a  small  error  in  latitude 
will  not  have  the  same  value  when  the  method  of  time-azimuth  is  employed  that  it  will  have 
in  the  altitude-azimuth  computation — excepting  as  stated  in  art.  13,  for  one  point  of  obser- 
vation of  the  celestial  body ;  namely,  when  either  q  or  Z  equals  90°,  whichever  may  be 
possible ;  at  e  or  /  the  numerical  values  of  the  resulting  errors  will  be  the  same  by  both 
methods,  but  the  signs  will  be  contrary.  Therefore,  in  practice,  having  determined  the 
azimuth,  and  afterwards  finding  that  the  latitude  used  is  slightly  erroneous,  care  must  be 
taken  in  correcting  the  result  that  dZ  is  not  computed  from  the  expression  derived  from 
the  formula  of  the  method  not  used  to  compute  the  azimuth.  The  foregoing  applies  as  well 
to  the  case  of  error  in  declination  ;  but  in  practice  the  occasions  will  be  few  when  the 
declination  can  be  regarded  as  having  any  appreciable  error.  For  discussion,  however,  a 
sensible  error  in  declination  affords  interesting  study,  and  it  will  not  be  omitted  in  showing 
the  relations  of  the  errors  in  the  data. 

Though  the  expressions  for  the  errors  are  not  strictly  true  excepting  when  the  incre- 

*  Except  in  the  one  peculiar  case  of  two  sides  and  the  angles  opposite  them  remaining  constantly  equal,  each  to 
90°;  while  the  remaining  side  and  its  opposite  angle  are  the  only  variables,  varying  by  the  same  amount,  being  always 
equal  each  to  the  other,  in  angular  measure. 


12  AZIMUTH. 

ments  of  change  are  infinitely  small,  yet  they  are  nearly  enough  true  provided  the  change 
in  circular  measure  is  sufficiently  small  to  be  regarded  equal  to  its  sine  ;  therefore,  since  an 
angle  that  does  not  exceed  one  degree  may  be  regarded  equal  to  its  sine  to  insure  accuracy 
to  the  nearest  second  in  the  result,  the  small  errors  in  the  data  usually  met  in  practice  may 
be  dealt  with.  dZ  may, "however,  fall  beyond  the  limits  allowable  as  a  finite  difference  for 
numerical  use  as  a  correction. 

26.  The  increment  of  change  is  really  length  in  arc.     If  given  in  seconds  or  minutes  it 
may  readily  be   reduced   to  circular  measure,  the  radius  being  unity ;  but,  in  practice,  this 
need  not  be  done,  for,  the  error  sought  being  expressed  in  the  same  unit  as  the  given  error 
(in  seconds  or  minutes),  the  factor  for  reduction  common  to  both  will  divide  out. 

27.  The  expressions  are  often  useful  to  find  the  actual  value  of  the  error  in  a  result  aris- 
ing from  the  employment  of  erroneous  data  ;   or,  speculatively,  to  determine  the  restrictions 
to  be  imposed  on  probable  errors  in  the  data,  in  order  to  keep  within  the  limits  of  error  allow- 
able in  the  result.   The  expressions  derived  are  for  errors;  hence  the  sign  must  be  changed  to 
make  them  corrections  to  be  applied  to  the  computed  azimuth. 

The  most  important  use,  however,  for  these  expressions  is  their  application  to  the  in- 
vestigation of  the  best  and  the  worst  conditions  for  observation  of  the  celestial  body. 

28.  For  convenience,  before  deducing  any  equation  of  maximum  and  minimum  errors  in 
the  computed  azimuth,  we  shall  derive  all  the  expressions  for  the  resulting  error  arising  from 
errors  in  the  data — taking  each   datum  separately  as  variable,  the  other  two  given  parts  as 
constant — in    the   several  methods  of  ascertaining  the  azimuth.     It  will  be  convenient   to 
adhere  to  the  familiar  common  notation  of  spherical  trigonometry — to  facilitate  the  changing 
of  expressions  into  others  equivalent — until  the  result  for  each  error  in  the  data  is  reached. 

Therefore  we  shall  have  with  the  astronomical  triangle  a  companion  triangle,  employing 
the  latter  to  obtain  the  partial  derivatives  (see  Figs.  I  and  2,  art.  18).  In  the  final  ex- 
pressions it  is  deemed  unnecessary  to  replace  dZ,  dh,  etc.,  by  AZ,  Ah,  etc.,  to  represent  finite 
differences. 

29.  I.  ALTITUDE-AZIMUTH. 
Given  h,  L,  and  d,  to  find  Z. 

1st  Case. — Error  in  Z  owing  to  error  in  h : 

Z  and  h  (B  and  d)  variable ; 
L  and  d  (c  and  b)  constant. 

cos  b  =  cos  c  cos  a  -|-  sin  c  sin  a  cos  B (25) 

o  =  —  cos  c  sin  ada-\-  sin  c  cos  a  cos  B  da  —  sin  c  sin  a  sin  B  dB ; 

sin  c  sin  a  sin  B  dB  =  —  (sin  a  cos  c  —  sin  c  cos  a  cos  B]  da  ; 

=  —  sin  b  cos  C  da ; 

,D  sin  b  cos  C 

art  = ; : : — =  da  ; 

sin  c  sin  a  sin  B 

,D  cot  C  sin  b       sin  c 

<*&  — = aa,  since  -: — ~  =  — — >=.; 

sin  a  sin  B      sin  C 


AZIMUTH.  13 

0  -  *)  ;   .      '       ' 


I.     2d  Case.  —  Error  in  Z  owing  to  error  in  L  : 

Z  and  L  (B  and  c)  variable  ; 
h  and  d  (a  and  b}  constant. 

By  interchanging  c  and  a  in  the  preceding  case  we  derive,  in  the  same  way  as  in  the  ist 
case, 

dL  .......     (27) 


^ 

tan  t  cos  L 

I.     ^d  Case.  —  Error  in  Z  owing  to  error  in  d\ 

Z  and  d  (B  and  b)  variable  ; 

h  and  L  (a  and  c]  constant. 
From  (25),  —  sin  b  db  =  —  sin  c  sin  a  sin  B  dB    .........     (28) 

JD  sin   b  J, 

dB  =  -.  ----  :  --  :  —  5  do  ; 
sin  c  sin  a  sin  B 

sin  b        sin  a 
.'.  since   -.  —  ~  —  —  —  -*, 
sin  B       sin  A 


.         . 
sin  c  sin  A 


--  j—.   - 

cos  L  sin  t 


A  —  —  ^ 

cos  Z,  sm  t 

Total  error,  dZ  =  dZK  -f-  dZL  -\-  dZ&  ; 


^—  :  dd  ............     (.29) 


dh  +  -  —  T1  -  -  dL  -  -^--l  -    dd     .....     (30) 


-  -  --  T  -  —  T  -  -T        -  -^--  - 

tan  q  cos  h  tan  t  cos  L  sm  /  cos  L 


14  AZIMUTH. 

30.  II.  TIME-AZIMUTH. 

Given  t,  d,  and  L,  to  find  Z. 

1st  Case.  —  Error  in  Z  owing  to  error  in  t: 

Z  and  t  (B  and  A)  variable  ; 
L  and  d  (c  and  b)  constant. 

sin  A  cot  B  —  sin  c  cot  b  —  cos  ccos  A  .........     (31) 

cos  A  cot  B  dA  —  sin  A  cosec"  B  dB  =  cos  c  sin  ^4  a^4  ; 
multiplying  by  sin  B, 

cos  A  cos  ^  «^4  —  sin  A  cosec  .5  dB  =  cos  £  sin  A  sin  .#  «L4  ; 

cos  A  cos  ^  —  cos  c  sin  ^4  sin  B  , 
dB  —  -      -  :  —  -f—       —^  --  dA  ; 
sin  A  cosec  B 

cos  C 
dB  —  --  : 


sm  A  cosec  B 

cos  C  sin  .5  .  cos  C  sin 


, 
dA  ; 


.  - 

sin  A  sin 

cos  ^  cos  d 
dZt  =  —  —.  —  dt  ...............     (32) 

cos  h 

II.     2d  Case.  —  Error  in  Z  owing  to  error  in  L: 

Z  and  L  (B  and  c]  variable  ; 

/  and  d  (A  and  b)  constant. 
From  (31),  -  sin  A  cosec2  BdB  =  cos  c  cot  b  dc  +  sin  c  cos  A  dc  ; 

•„   A  i  DJD       cos  c  cos  b  -4-  sin  <£  sin  c  cos  y2  , 

—  sin  A  cosec  Z&/Z?  =  --  !  --  -  --  dc  ' 

sm  <7 

sin  A  cos  # 

~slr?"^^:  =  lirr^^;    ..........     (33) 


sin*       sin* 

sin^       sin* 


trig.,  . 

'     ...........     (34) 


AZIMUTH.  15 

.  • .  multiplying  the  first  and  second  members  of  (33)  by  the  first  and  second  members  of  (34), 
respectively, 

dB  =  cot  a  dc ; 


sin  B 

dB  =  —  cot  a  sin  B  dc  ; 
dZ  =  —  tan  //  sin  Zd  (go  —  L)  ; 
dZL  =  tan  h  sin  ZdL  ..............     (35) 

II.     $d  Case.  —  Error  in  Z  owing  to  error  in  d\ 

Z  and  d  (B  and  b)  variable  ; 
L  and  /  (c  and  A)  constant. 
From  (31),  —  sin  A  cosec2  B  dB  =  —  sin  c  cosec2  b  db\ 

multiplying  by  sin  B  sin  b, 

sin  A  sin  b  sin  c  sin  B 

-  .  —  5  —  dB  =  -  :  —  7  —  do  ; 
sin  B  sin  b 

sin  A       sin  a  sin  c       sin  C 

by  trig.,  -  —  =  =  ~^—  /<         and         -:  —  -,  =  -.  —  D  ; 

sin  ^       sin  0  sin  b       sin  /?  ' 


/.  sin  0d!Z?  —  sin  C  db  ; 

cos  ^  dZ  =  sin  ^</  (90°  —  d}  ; 

dZ*  -  -  -—  ,  dd.  (36) 

cos  h 

Total  error,  </Z  =  dZt  -\-  dZL  -\-  dZd. 

cos  a  cos  df  ,  sin  a 

dZ=  --        —.  —  dt  4-  tan  h  sin  ZdL  ---  -.  dd.       .....     (37) 

COS  IL  COS  k 

31.  III.    TIME-ALTITUDE-AZIMUTH. 

Given  h,  t,  and  A,  to  find  7,. 

1st  Case.  —  Error  in  Z  owing  to  error  in  h  : 

Z  and  h  (B  and  a)  variable  ; 
/  and  d  (A  and  b)  constant. 

sin  A  sin  b 
sin  B  =  -  -  ;      ..........     (38) 

sin  a  w  ' 


i6 


cos  B  dB  =  — 


AZIMUTH. 

sin  y4  sin  h  cos  # 


sin 


•da; 


sn 


sin     cos  a 


III. 


From  (38), 


cos  B  dB  =  - 

dB=- 
dZ  =  - 
dZh  = 


sin  b 


X  sin  b  cot  a  da  ; 


tan  B  cot  #  <afo  ; 
ten  Z  tan  /;  ^(90°  — 
tean  Z  ta.n  h  dh.      . 


III.     3</  CVw.  — 


From  (38), 


Total  error, 


-Error  in  Z  owing  to  error  in  t : 

Z  and  t  (A  and  B)  variable  ; 
d  and  h  (b  and  a)  constant. 

cos  B  dB  —  cos  A  sin  b  cosec  a  dA 

cos  A  sin  B 
sin  A 


dB  =  cot  y4  tan  B  dA  ; 
dZ,  =  cot  /  tan  Z  dt. 
Error  in  Z  owing  to  error  in  d: 


Z  and  d  (B  and  b}  variable  ; 
/  and  h  (A  and  a)  constant. 


sin  A 

cos  B  dB  =  —. —  cos  b  do ; 
sm  a 


sin  B 

cos  B  dB  =  —. — T  cos  b  db  ; 
sin  b 


dB  =  tan  B  cot  b  db\ 
dZ  =  tan  Z  tan 
dZ   =  —  tan  Ztan 


=  tan  Z  tan  h  dh  -j-  tan  Z  cot  t  dt  —  tan  Z  tan 


32.  IV.    HORIZON-AZIMUTH. 

(A  special  case  of  altitude-azimuth,  in  which  h  =  o) 
Given  h  =  o,  L,  ««^  d,  /0  yf«^  Z. 
U/  Case—  Error  in  Z  owing  to  error  in  L  : 


(39) 


(40) 


(40 
(42) 


Mr  \ 

HtJNIYEKJ  "IV  \ 
AZIMUTH. 

Z  and  L  variable  ; 
^and  */ constant. 


By  (27), 


I 


For  another  form,  since  /z  =  o,         tan  t  = 


tan  /  cos  / 
tan  Z 


—TdL. 


sm  Z.' 
z  =  —  tan  Z  cot  Z  dL. 


IV.     2</  Case.  —  Error  in  Z  owing  to  error  in  d: 

Z  and  d  variable  ; 
h  and  L  constant. 


By  (29), 

For  another  form,  since  h  =  o, 


sin  /  cos  L 

sin  q 
sin  t  =  — 

cosZ- 

dZd  =  —  cosec  q  dd. 


dd. 


(44)  and  (46)  rnay  also  be  found  from 
the  right  spherical  triangle  PNO,  by  differ- 
entiating sin  d  =  cos  Z  cos  Z,  since  the  right 
angle  formed  by  the  intersection  of  the  me- 
ridian and  the  horizon  is  constant,  and  for 
another  form  to  (46)  we  have 

dZd  =  —  cot  Z  cot  d  dd. 


(IV.) 

Given  L  and  d,  and  PNO  =  90°,  to  find  Z. 
1st  Case. — Error  in  Z  owing  to  error  in  L  : 


FIG.  3. 


Z  and  L  variable  ; 

PNO  =  90°,  and  d  constant. 


cos  Z  — 


sin  d 
cos  L' 


. 
—  sm  Z  dZ  = 


sin 


in  L 
—  dL  ; 


—  sin  Z  dZ  —  cos  Z 


cosa  L 
sin  L 


'dL; 


cos  L 
dZL  =  —  tan  L  cot  Z  dL  (44). 


(43) 


(44) 


(45) 


(46) 


PN-L\ 


P0=  p- 
NOP  =  90°  -  q  ; 
/W0  =  90°. 


(47) 


(48) 


AZIMUTH. 


(IV.)    2d  Case. — Error  in  Z  owing  to  error  in  d: 


Z  and  */  variable  ; 

PNO  =  90°,  and  L  constant. 


From  (47), 


-  sin  Z  dZ  = 


cos  d  sin  Z 


dZd  =  —  cosec  q  dd  .  .  .  (46)  ..........     (49) 

tan  Z 
In  the  triangle  PNO,  cos  (90°  —  q)  = 

tan  Z 

.  '  .  sin  q  =  —  —  >  ; 
cot  d 

.  '  .  from  (49),  dZd  —  —  cot  d  cot  Z  dd.  .............     (50) 

Total  error  dZ  =  —  tan  L  cot  Z  dL  —  cot  </cot  Z<W.    ......     ($i) 

33.     The  following  formulas  will  sometimes  be  found  useful  for  purposes  of  elimination, 
and  reduction  of  equations  that  occur  subsequently. 

By  interchanging  d  and  L,  Z  and  q  ;  d  and  L  being  constant,  we  derive,  similarly  to  (26), 

dqk  =  i  -  ^  ---  fdh  ;     ...........     (52) 

tan  Z  cos  # 

cos  Z  cos  Z  . 
and  similarly  to  (32),  dqt  =  --      —  y  -  at  ............     (53) 

COS   rt  < 

,  sin  q  cos  a 

Also  from  sin  Z  =  -  —  P  —  ; 

cos  L 

T          J    J  rr    T7          COS  d  COS  0  r 

when  L  and  <z  are  constant,      cos  Z  dZ  =  —     —r—da  : 

cos  L       *  ' 

cos  */    cos  ^ 
My  =  --  =•  .  --  \,dq  ...........     (54) 

cos  L     cos  Z  *  v:^' 


(54)  may  also  be  obtained  by  eliminating  dh  from  (26)  and  (52);  or  by  eliminating  dt  from 
(32)  and  (53). 


PART  IV. 

CONSIDERATIONS  AFFECTING  THE  EQUATIONS  OF  MAXI- 
MUM AND  MINIMUM  ERRORS,  AND  RESPECTING  THE 
CURVES  OF  THESE  ERRORS. 


34.  Though  in  computing  the  azimuth  it  is  convenient  to  regard  all  the  angles  as  positive 
within  the  limit  of  180°  (see  art.  17),  on  whichever  side  of  the  meridian  the  body  is  observed, 
yet  looking  to  the  meridian  for  a  possible  locus  of  algebraic  maximum  and  minimum  errors 
occurring — to  determine  the  truth  by  inspection  —  it  will  be  necessary  to  follow  the  star  in  its 
diurnal  course  and,  therefore,  to  consider  the  general  astronomical  triangle;  necessary,  also, 
in  order  to  discriminate  analytically  the  maximum  and  minimum  errors  wherever  occurring 
in  the  star's  path.  Following  this  course,  /  will  be  reckoned  from  o°,  at  the  upper  culmina- 
tion of  the  body  to  the  westward  to  360°.  Z  will  be  reckoned  from  the  point  of  the  horizon 
nearest  the  elevated  pole  to  the  westward  around  to  360°,  as  follows: 

If  -f-  d  >  L,  at  the  upper  transit  on  the  meridian  Z  passes  through  o°,  from  Z  <  360  to  Z  > 

o ;  increases  to  a  maximum  value  <  90°,  when  q  =  90,  then  decreases  to  \    f-0  >  at  lower  transit ; 

(  3°°    ) 

from  360°,  decreases  to  a  minimum  value  >  270°,  when  q  =  270,  then  increases  to  j  ^  0    [at 

upper  transit. 

If  ±  d  <  L,  Z  at  upper  transit  passes  through  180°,  from^>  180°  to  Z  <,  180°;  de- 
creases through  90°  to  \  ^  o  f  at  lower  transit  ;  from  360°  decreases  through  270°  to  180° 

(  3°°    ) 

at  upper  transit. 

If  —  d^>  L,  Z  at  upper  transit  passes  through  180°,  from  ^>  180°  to  2T<  180°;  de 
creases  to  a  minimum  >  90°,  when  q  =  90°,  then  increases  to  180°  at  lower  transit  ;  increases 
to  a  maximum  <  270°,  when  q  =  270°,  then  decreases  to  180°  at  upper  transit. 

q  will  be  reckoned  to  a  possible  360°  as  follows :  If  -f-  af>  L,  q  =  180°  at   upper   transit ; 

decreases  through  90°  to     •<     ,-  0  [     at  lower  transit ;  from  360°  decreases  through  270°  to 

(  3°°    ) 

1 80°  at  upper  transit. 

If  ±  d  <  L,  q  —  o  at  upper  transit ;  attains  a  maximum  value  <  90°,  when  Z—  90°  then 

decreases  to   \    ,-  0  [at   lower  transit;  from   360°  decreases   to   a  minimum   >  270°,  when 
;      (  3°°    > 

Z  =  270°  then  increases  to    j  3  at  upper  transit. 

If  —  d>  L,    q  —  o  at  upper  transit,    increases,    passing  through  90°,  to  180°   at  lower 

(  360°  ) 
transit ;  continues  to  increase  through  270°  to  j  •*  0    [    at  upper  transit. 

Briefly,  whenever   the  given  star  crosses  the  meridian  of  a  given  place,  every  one  of  the 


20  AZIMUTH. 

angles  /,  Z,  and  q  passes  through  either  o°  or  180°;  rot  necessarily  the  same  value  for  all  at 
the  same  time.  But  each  angle  is  <  180°  or  >  180°  according  as  the  other  angles  are,  each, 
<  180°  or  >  180°;  the  sides  remaining  as  noted  in  article  16. 

35.  For  use  in  a  given  observation  the  higher  equations  giving  points  of  maximum  and 
minimum  errors  will  be  made  to  consist  of  terms  in  some  trigonometric  functions  of  L  and 
d  and  some  one  function  of  /  (or  of  h}.     Substituting  the  values  of  L  and  d  that  belong  to 
the  particular  occasion  for  finding  the  azimuth,  /  (or  h}  can  be  determined,  to  know  when  to 
observe  the  body  so  that  the  resulting  error  shall  be  a  minimum ;  or,  to  know  what  point 
of  observation — and  its  neighborhood — to  avoid  as  giving  a  maximum  error. 

For  convenience  in  tracing  the  locus  of  the  equation,  and  for  the  best  exhibition  of  it,  L 
will  be  regarded  as  fixed,  with  d  and  /  (or  h)  variable :  hence,  giving  different  values  to  d, 
those  of  t  (or  h},  corresponding,  will  be  found. 

36.  In  the  equation  to  the  locus,  with  given  values  of  L  and  d,  and  calling,  for  instance, 
sin  h  =  x,  the  roots  found  from  the  numerical  equation  embracing  both  real  and  imaginary 
ones,  even  though  every  real  root  correspond  to  some  point  of  a  plane  curve  regarded  as 
purely  algebraic,  in  which  x  may  have  any  value,  not  being  restricted  to  the  value  of  sin  h 
lying  between  -f-  I  and  —  I,  yet  the  locus  on  the  sphere  may  not  contain  every  point  repre- 
sented by  the  real  roots.     For  if  one  of  these  gives  sin  h  >  ±  I,  it  will  present  an  impossible 
case.     Again,  the  root  may  have  a  value  not  impossible  for  the  trigonometric  function,  yet 
one  that  sin  h  never  attains.     For  illustration,  when  d=  o  the  greatest  value  that  h  can  have 
is  ±  (90° —  Z),  when  sin  h  =  ±  cos  Z;  hence,  in  this  case  should  the  root  have  a  numerical 
value  greater  than  cos  L  it  will  be  inadmissible. 

Therefore,  among  several  real  roots  there  may  be  but  one  value  that  is  admissible.  The 
problem  may  also  be  materially  simplified  on  account  of  positions  of  the  body  that  are 
impracticable  for  observation  ;  for  example,  the  diurnal  circle  may  cross  a  curve  of  minimum 
errors  below  the  horizon ;  then  (theoretically)  the  best  practicable  position  of  the  body  is  in 
the  horizon. 

37.  While  equations  in  the  terms  mentioned  must  be  resorted  to  for  finding  the  time  or 
the  altitude  at  which  most  favorably  to  observe  a  given  body  in  the  observer's  latitude,  and 
though  incidentally  useful  in  constructing  the  curves  of  maximum  and  minimum  errors,  yet, 
for  the  latter  purpose,  equations  in   terms   of  Z,  L,  and  h  are  far  simpler.*     By  them,  the 
point  tracing  the  locus  is  virtually  referred  to  polar  co-ordinates  instead  of  the  more  compli- 
cated bi-polar  co-ordinates  employed  with  the  equations  in  terms  of  t,  Z,  d,  and  of  //,  Z,  d. 

But,  more  than  this,  the  system  of  polar  spherical  co-ordinates  may  readily  be  trans- 
formed into  a  system  of  plane  rectangular  co-ordinates  for  the  projection  of  the  curve.  By 
employing  the  equations  to  the  latter,  the  difficulties  inherent  in  all  the  other  equations  for 
determining  all  the  properties  of  the  curve  are  removed,  and  analysis  may  be  made.  With- 
out a  knowledge  of  the  properties  of  the  curve  of  projection,  a  true  conception  of  the  curve 
on  the  surface  of  the  sphere  is  unattainable. 

There  remain  two  other  forms  of  equations  that  are  interesting  and  incidentally  useful  ; 
namely,  (i)  the  equation  referring  the  point  to  the  system  of  rectangular  spherical  co-ordi- 
nates, the  origin  being  the  zenith,  and  the  spherical  axes  the  meridian  and  prime  vertical ; 
(2)  the  polar  equation  to  the  plane  curve  of  the  projection  in  which  6  is  the  complement  of 
the  azimuth  and  r  the  linear  distance  corresponding  to  the  projection  of  the  zenith  distance. 

38.  The  diagrams  are  stereographic  projections  on  the  primitive  plane  of  the  horizon, 
*  Insisting  on  the  use  of  the  horizon  as  the  primitive  plane,  for  the  best  view  of  the  curve  (article  38). 


AZIMUTH.  21 

the  point  of  sight  being  at  the  nadir.  The  parts  lying  above  the  horizon  are  drawn  in  full 
lines  ;  those  below  the  horizon,  in  broken  lines.  The  distortion  in  the  projection  of  the 
lower  hemisphere  gives  an  appearance  of  asymptotic  properties  to  branches  of  the  curves 
passing  through  the  nadir,  and  of  lack  of  symmetry  in  the  branches  of  the  loci  above  and 
below  the  horizon.  But  symmetry  exists,  and  may  be  more  easily  perceived  if  the  lower 
hemisphere  is  revolved  180°  about  a  tangent  at  the  east  point,  the  point  of  sight  being 
moved  to  the  new  pole  of  the  primitive  circle.  This  can  be  easily  sketched  without  attempt 
at  great  accuracy  ;  but  the  stereographic  projection  is  considered  preferable  for  the  cases 
presented.  In  the  projection  itself  the  asymptotic  properties  do  exist. 

No  attempt  is  made  to  construct  diagrams  on  a  sufficient  scale,  or  with  the  precision 
necessary,  for  use  in  graphically  finding  the  point  to  accept  as  the  best,  or  to  avoid  as  the 
worst,  in  observing  the  body. 

39.  Tables  should  be  prepared  giving  the  most   favorable  time  for  observation,  when 
error  in  h  in  the  altitude-azimuth  and   error  in  /  in  the  time-azimuth  are  considered,  in  the 
case  of  bodies  that  cross  the  prime-vertical  in  their  diurnal  course  ;  and  tables  to  give  the 
dividing   line   between    altitude-azimuth   and    time-azimuth  being  the  better  method  when 
error  in  latitude  alone  is  concerned.     Tables  applicable  to  other  cases  would  be  useful,  but 
they  are  not  so  much  needed  as  are  those  specified. 

40.  Since  it  is  purposed  to  determine,  from  investigation  of  the  expression  for  error,  the 
best  and  the  worst  conditions  for  observation — numerical  maxima  and  numerical  minima,  ir- 
respective of  sign,  will  often  be  called  max.  and  min.  ;  notwithstanding  that  when  found  alge- 
braically the   numerical  max.  may  be  an  algebraic   min.,  and  the  numerical  min.  an  algebraic 
max.,  according  to  the  sign.     In  general,  in    this   treatise,  if  the   terms  maximum  error  and 
minimum  error  occur   unqualified  they  will  mean  numerically  greatest   and   least  errors,  re- 
spectively. 

Algebraic  max.  and  min.  corresponding  to  true  max.  and  min.  where  true  is  the  term 
used  by  mathematicians  respecting  max.  and  min.  found  by  giving  to  the  first  differential  co- 
efficient the  value  zero. 

Inspection  of  the  given  expression  will  usually  show  whether  it  is  susceptible  of  algebraic 
max.  and  min.  If  these  exist  it  will  sometimes  further  be  seen  at  exactly  what  position  of 
the  heavenly  body  they  occur.  If  not  obvious,  recourse  must  be  had  to  the  analytical  method 
of  finding  the  precise  conditions  giving  them. 

But  inspection  may  show  that,  though  no  true  max.  or  min.  exists,  yet  values  of  o  and 
oo  do  occur,  and,  for  our  purpose,  these  are  the  truest  maxima  and  minima, — sacrificing  English 
to  emphasis.  To  distinguish  these  from  true  (algebraic)  max.  and  min.,  and  from  finite  nu- 
merical max..  and  min.,  the  term  absolute  will  be  used  in  this  treatise  ;  dZ  =  o,  an  absolute 
min.;  d Z '=  oo  ,  an  absolute  max. 

41.  To  facilitate  comparisons  to  be  made  hereafter,  it  will  be  convenient  to  collect  the 
several  expressions  for  errors  and  to  give  to  each  one  some  equivalent  forms  from  trigonom- 
etry not  given  in  what  precedes. 


22  AZIMUTH. 

I.    ALTITUDE-AZIMUTH. 

dZ  I  cos  q  cos  ^ 

No.  I.     -77-  =        -  --  7  =  —  --  7  =  —.  —        —f  ;  L  and  <z  constant  (20). 
tf/z  tan      cos  ^       sin      cos  ^       sin  /  cos  L 


No.  2.     -77-  = 


dZ  I  cos  /  cos 


—          7-       -  — 
tan  t  cos  Z       sin  /  cos  L      sin      cos 


j  ;  h  and  #  constant  (27). 
h  v  '' 


dZ  i  i  cos  d 

No'  3'     dd  =  ~  shTTcoIZ  =     ~  iiHT^A  =  "  sin  Z  cos  A  cos  Z  ;  h  and  Z  Constant  (29)-  (57) 


II.    TIME-AZIMUTH. 
cos  q  sin  Z  cos  a  cos  d  cos  d  sin  a  cos  a 


<£?  sin  ^  sin  Z      sin  /£  sin  ^  sin  q 

No.  5.     -77  =       tan  h  sin  Z=  -       —  7-  —  =  -  :  —        —=  —  = 
dL  cos  h  sin  ^  cos  L 


tan  //  sin  ^  cos  d 

-^TZr~    ~;  ^and^  constant  (35).   .     (59) 

«J^  sin  q  sin  /  cos  L  sin2  #•  sin  ^  sin  Z 

^Q    ^        -  —  __  i.  —  _  _  —   _  .  _  ±___  —   _  _  ±  __ 

dd  cosk  cos2  h  sin  /  cos  L  sin  /  cos  d 

sin  ^  cos  L  sin2  .Zcos  Z 

'  ^Xc~o7^=  "  lET^Trf  ;  '  and  L  constant  (36).  (60) 

Inspection  of  (55),  (58),  and  of  (57),  (60),  shows  the  truth  of  the  last  paragraph  of  article  12. 

III.      TlME-ALT.-AZIMUTH. 

No.7.     d4=       tan  Z  tan  h  =  J^j".  9  ^  h  =  smj_cosd  sin  h 
<*H  sm  t  cos  L  cos  Zcos  Z  cos  k 

sin  /  cos  </  sin  h 
cos2  4  cos  Z    ;  '  and  ^  constant  (39)-     (61) 

M      Q      dZ  sin  Z  cos  /       cos  d  cos  / 

No.  8.     -77  =       tan  Z  cot  /  =  --  =,—  —  -  =  --  =,— 

at  cos  Z,  sin  /       cos  Z  cos  k 

cos  </  cos  /  sin  ^ 
=  c^Z^oIZ^hT/  ''  h  and  ^constant  (40).     (62) 

No.  9.  =-ta  sin 


.  _ 

cos  Z  cos  Z  cos  Z  cos  /* 

tan  Z  sin  /  sin  d 


-     (63) 


AZIMUTH.  23 

Useful  expressions  when  d  and  L  are  constant. 

I  cos  Z  cos  Z  cos  L  cos  Z 


dh  ~ 

tan  Z  cos  h 

sin  Zcos  h      sin  £ 

cos  d       s 

;in  ^  cos  /z 

—  T  iS2).        .       . 

cos  d  VJ  y 

.     .     .     ^04j 

dq 

cos  Z  cos  L 

cos  Z  sin  ^ 

cos  Z  sin 

Z"  cos  Z 

cos  df  sin  ^   /(< 

•»\                     /Ar\ 

dt 

cos  h 

sin  £ 

sin  t 

cos  d 

cos  h  tan  Z"  ^ 

3)-    •       •       (05) 

dZ 

cos  q  cos  df 

cos  q  sin  Z 

rnf  . 

cos  ^  cos 

q  tan  Z 

1C  11 

(fff\ 

Hn 

mi  -/f  rnQ   A 

ro«  X  «;in  /7 

sin  /  r 

n«  T.         w  '     ' 

'       '       '       \P°) 

42.  In  deriving  each  of  the  differential  coefficients  in  what  precedes,  it  was  deemed  ad- 
visable, in  the  first  instance,  for  the  purposes  of  this  work,  to  use  the  fundamental  equation 
involving  the  four  parts  to  be  considered  and  thus  find  the  partial  differential  required. 

But  for  the  multiplicity  of  cases  of  differential  variations  of  the  astronomical  triangle 
that  may  arise,  the  work  may  be  facilitated  by  employing  the  three  equations  (73),  (74),  and 
(75),  given  below,  derived  as  follows  : 

By  spherical  trigonometry, 

cos  a  =  cos  b  cos  £  -f-  sin  b  sin  c  cos  A  ;  \ 

cos  b  =  cos  a  cos  c  -f-  sin  a  sin  £  cos  B  ;  >  .......     (67) 

cos  ^  =  cos  a  cos  b  -\-  sin  a  sin  b  cos  C.  > 

Differentiating  the  first  of  (67),  all  parts  being  variable, 

—  sin  a  da  =  —  sin  b  cos  £  db  —  cos  b  sin  cdc  -f-  cos  b  sin  c  cos  A  db 

-f-sin  b  cose  cos  A  dc  —  sin  £sin  csin  A  dA.  .     (68) 


{—  (sin  b  cos  t—  cos  b  sin  ccosyi)  d#  \          /  —  sinacosCdb  \ 

—  (sin  ^cos  #  —  cos  £  sin  dcosA)dc  >•   =   •<  —  sin  #  cos  ,#  afc  V.     .     (69) 

—  sin  b  sin  cs'm  A  dA  '  —  sin  a  sin  b  sin 


Dividing  out  sin  a,       —  da  =  —  cos  C  db  —  cos  B  dc  —  sin  b  sin  £"<aL4  .......  (70) 

In  the  same  way  from  the  second  and  third  of  (67),  we  have 

—  db  =  —  cos  A  dc  —  cos  Cda  —  sin  ^  sin  A  dB  .......  (71) 

—  dc  =  —  cos  B  da  —  cos  A  db  —  sin  a  sin  B  dC.      ......  (72) 

Whence,  substituting  angles  and  complements  of  the  sides  in  the  astronomical  triangle, 

dh  =  cosqdd-\-  cos  Z  dL  —  cosdsinqdt  ........  (73) 

dd  =•  costdL-{-  cosqdh  —  cosL  sin  t  dZ.     .......  (74) 

dL  =  cos  Zdh  -J-  cos  /  dd  —  coshs'mZdg  ........  (75) 


AZIMUTH. 


Since,  in  obtaining  the  partial  differential  used,  we  consider  two  parts  of  the  triangle 
constant  in  any  given  case,  it  will  require  not  more  than  two  of  the  equations  (70),  (71),  (72), 
or  of  (73),  (74),  (75),  to  obtain  any  one  of  the  differential  coefficients. 

From  (73),  (74),  (75),  some  one  of  the  expressions  in  each  group  of  (55)  to  (66)  will  be 
derived  directly. 

To  obtain  other  equivalent  expressions,  the  following  fundamental  formulas  are  often 
required : 


sina^-f-  cos8.*-  =  I  ; 

cos  A  =  —  cos  B  cos  C-\-  sin  B  sin  C  cos  # ; 
cos  B  =  —  cos  C"  cos  .4  -|-  sin  dTsin  y2  cos  #  ; 
cos  C  =  —  cos  A  cos  .5  -|~  sm  -A  sin  .Z?  cos  ^ ; 
cos/   = — cos  ^  cos  ^  -f-  sin^sin  ^sin  h\ 
cos  Z  =  —  cos  ^  cos  /  -f-  sin  q  sit  /  sin  d ; 
cos  q  =  —  cos  /  cos  Z  -J-  sin  /  sin  ^  sin  Z. 


PART  V. 

DETERMINATION,  BY  INSPECTION,  OF  THE  MOST  FAVOR- 
ABLE  AND  THE  LEAST  FAVORABLE  POSITIONS  OF  A 
GIVEN  BODY  FOR  OBSERVATION  IN  A  GIVEN  LATITUDE. 


43.  I.  ALTITUDE-AZIMUTH. 

No.  i.  From  (55)  it  is  seen  that  for  error  in  altitude  an  absolute  max.  occurs  on  the 
meridian  for  all  bodies,  q  =  o  or  180°  :  an  absolute  min.,  when  q  =  90°  or  270°  ;  hence,  for 
each  of  all  bodies  having  ±  d  >  L  at  its  elongation ;  but,  for  these  bodies,  no  algebraic  max. 
or  min. 

For  all  bodies  whose  d  <  ±  Z,  algebraic  max.  and  min.  occur,  both  being  numerical 
min.  Where  the  min.  occurs  is  not  obvious,  though  text-writers  have  given  overwhelming 
preponderance  to  tan  q,  treating  cos  h  as  insignificant,  and  thus  have  erroneously  assigned 
the  min.  error  to  the  position  on  the  prime-vertical.  (See  articles  5  to  9.) 

In  the  case  of  ±  d  >  Z,  when  Z '=  90°  is  never  attained,  incidentally  the  min.  occurs 
when  the  body  is  nearest  in  azimuth  to  the  prime-vertical.  But  for  ±  d  <  L  recourse  must  be 

had  to  solving  d\- r]  =  o  to  obtain  the  conditions  giving  algebraic   max.  and  min., 

\tan  q  cos  h] 

which  are  numerical  min.  Though  the  exact  point  desired  (on  either  side  of  the  meridian)  is 
not  obvious,  yet  it  is  seen  to  lie  on  that  side  of  the  prime-vertical  towards  the  nearer  pole, 
since  for  equal  values  of  q  on  each  side  of  the  prime-vertical  ±  h  will  there  be  less  and  cos  h 
greater.  This  branch  of  the  curve  of  min.  errors — from  zenith  to  nadir  through  the  east  and 
west  points — together  with  the  branch  derived  from  q  =  90,  giving  the  entire  curve  of  min. 
errors,  is  shown  in  locus  No.  I,  the  meridian  being  the  locus  of  max.  errors. 

It  will  readily  be  seen  that  the  text-writers  would  have  erred  to  a  less  degree  had  they 
assigned  the  most  favorable  position  to  the  six-hour  circle  instead  of  to  the  prime-vertical. 
For,  if  d  <  L, 


26  AZIMUTH. 


when  Z  =  goc 


tan  q  = 


cot  L 
cos  h' 


tan  q  =  -         , 
cos  d 

and  (55)  becomes 


dZ 

—TT  =  tan  L.      .     .     .     (tf) 
dh 


When  /  =  90° 


cot  L 


tan  Z  cos 


=  tan  L  sin  Z.  .     .     .     (b) 


(b)  <  (#)  excepting  for  d  =  o°,  when  /  =  90°  and  Z  =  90°  occur  at  the  same  time.     (See 
arts.  5  and  15.) 

44.  The  following  brief  memoranda  result  from  an  inspection  of  the  several  coefficients 
of  error,  L  and  d  being  now  constant,  even  though  the  coefficient  is  that  for  error,  or  change, 
in  one  of  them  ;  hence  that  one  a  variable  in  getting  the  expression  for  error. 

No.  2.     From  ($6),      -,-=r  = =  ,  for  error  in  L,  absolute  max.  when  t  =  o°  or  180°  ; 

3  '      dL       tan  t  cos  L 

absolute  min.  when  t  =  90°  or  270°.     No  algebraic  max.  or  min.     The  locus  of  max.  errors, 
the  meridian  ;  that  of  min.  errors,  the  six-hour  circle. 

No.  T,.     From   (^7),        —,=• : f,  for  error  in  d,  absolute  max.  when  t  =  o°  or 

dd  sin  t  cos  L 

1 80°;    numerical   min.  when   /  —  90°  or  270°.      Loci:  the  meridian  for  max.;  the  six-hour 
circle  for  min. 

The  numerical  min.  is  obvious,  but  may  be  verified  by  finding  the  algebraic  max.  and 
min.  The  loci  being  the  same  as  for  error  in  L  (No.  2),  but  the  min.  error  >  o. 


45.  II.  TIME-AZIMUTH. 

dZ      —  cos  q  cos  d 

No.  4.     From  (58),        -,-  = ~ ,  for  error  in  t. 

dt  cos  h 

For  ±  d  >  L,  absolute  min.  when  q  =  90°,  270°. 

For  ±  d  <  L,  numerical  min.  erroneously  assigned  to  the  prime-vertical  by  some  text- 
writers. 

For  all  bodies  algebraic  max.  and  min.  generally  giving  numerical  max.  when  q  =  o,  180°. 

Solving  d  f-      .       — j  =  o,  algebraic  max.  and  min.  will  be  found,  giving  numerical  max. 

generally  when  q  =  o,  180°  ;  and  for  d  <  L  numerical  min.,  which,  though  the  point  is  not 
obvious,  proves  to  lie  on  that  side  of  the  prime-vertical  towards  the  nearer  pole.  In  very 
high  latitudes,  for  some  declinations,  this  point  is  very  near  the  meridian  and  very  far  from 
the  prime-vertical,  in  bearing;  hence  the  importance  to  arctic  explorers  to  know  the  truth, 
not  to  be  misled  by  false  teaching. 

As  in  No.  i  (see  last  paragraph  of  art.  43)  the  prime-vertical  gives  a  less  favorable  posi- 
tion of  the  body  than  that  on  the  six-hour  circle.     For  if  d  <  Z, 


AZIMUTH. 


when  Z  —  90°,  When  t  =  90°, 

dZ  cos,  d  •'•  dZ  cos 


=  COS  Q r  77=  COS 


7,       —~~     V*WO     I/  »  7,      \^V/^»    */  7 

at  cos  #  at  *  cos  # 

tan  af    cos  ^  cot</  cos  a? 


tan  ^  '  cos  ^  cot  /i  '  cos  /£ 

sin  d  _  cos2  a?  sin  h 

~"  sin  ^  ~  cos2  h '  sin  d 

=  sin  Z.  (V)                                          cos8  d 


cos' 
sin2/ 
sin2  z 


. 
sin  L 


sn 


Excepting  when  d=  o,  t  =  Z  =  90°,  (d}  is  always  less  than  (c)  ;  for  when  t  —  90°,  z  is  nearer 
to  90°  than  is/. 

sin2/  . 

. ' .  -7—5 —  is  a  proper  fraction, 
sin  # 

Loci :  Absolute  min.,  curve  of  q  =  90°.     Numerical  min.,  curve  shown  in  No.  4.     Nu- 
merical max.,  the  meridian. 

No.  5.     From  (59),  -jj  =  tan  h  sin  Z,  for  error  in  L.     Absolute  min.,  when  h  =  o°,  also 

when  Z  =0°,  180°.  For  rising-and-setting  bodies  numerical  max.  in  each  quadrant ;  for  a  cir- 
cumpolar  star,  a  max.  on  each  side  of  the  meridian,  above  the  horizon  ;  for  a  star  always 
below  the  horizon,  a  max.  on  each  side  of  the  meridian.  The  positions  giving  numerical 
max.  not  obvious,  but  corresponding  to  algebraic  max.  and  min.  found  from  d(ia.n  h  sin  Z) 
=  o. 

Loci :  Absolute  min.,  the  horizon  and  the  meridian  ;  numerical  max.  on  curve  No.  5. 

7  ^ 

No.  6.     From  (60),  -jj  =• 7,  for  error  in  d.     Absolute  min.  when  q  —  o°,  180°  and  h 

J  dd  cosk 

not  90°.     Numerical  max.  and  min.  from  algebraic  max.  and  min.  found  from  d\—    -7)  =  o. 

\cos  hi 

Inspection  alone  does  not  serve  to  determine  even  approximately  all  the  positions  of  the 
body  causing  max.  and  min.  errors :  but  in  the  single  case  of  d  >  Z,  it  is  obvious  that  there 
is  one  max.  on  each  side  of  the  meridian,  occurring  between  the  times  of  the  upper  culmina- 
tion of  the  body  and  its  elongation  ;  since  for  equal  values  of  sin  q  on  each  side  of  unity 
(q  =  90),  cos  h  will  be  less  for  the  greater  altitude. 

Loci :  Absolute  min.,  the  meridian  ;  numerical  max.  and  min.  given  by  curve  No.  6. 


III.    TIME-ALTITUDE-AZIMUTH. 

7^ 

46.  No.  7.     From  (61),  —TV  =  tan  Z  tan  h,  for  error  in  h.     Absolute  max.  when^=  90°, 
270°.    Absolute  min.  when  Z  =  o°,  180°  also  when  k  =  o.     Numerical  max.  from  algebraic  max. 


28  AZIMUTH. 

and  min.  found  by  d(tan  Z  tan  h)  =  o.  There  will  he  a  max.,  then,  for  any  star  that  does  not 
cross  the  prime-vertical  (±  d>  Z,),  on  each  side  of  the  meridian  above  and  also  below  the 
horizon,  if  a  rising-and-setting  body.  'If  the  body  is  always  visible,  a  max.  above  the  horizon 
on  each  side  of  the  meridian  ;  if  always  hidden,  a  max.  on  each  side  of  the  merid.  below  the 
horizon.  For  bodies  that  cross  the  prime-vertical  above  the  horizon  (-f-  d  <  L),  a  max.  be- 
low the  horizon  on  each  side  of  the  meridian  ;  for  those  that  cross  below  (—  d  <  L),  a  max. 
above,  on  each  side. 

Loci  :  Absolute  max.,  the  prime-vertical  ;  absolute  min.,  the  horizon  and  the  meridian  ; 
numerical  max.  by  curve  No.  7. 

sf  "7          4-  *7 

No.  8.     From    (62),  -  r  =  -  ,  for   error   in   /.     Absolute  max.,  Z  =  00°  (when  /  is 
at        tan  r 

not  90°)  ;  absolute  min.,   t  =  90°  (when  Z  is  not  90°)  ;  numerical   max.  or  min.  when  Z  = 

/cos  d  cos  /\ 
o,  180°,  and   t  =  o,   180°,  from    algebraic    max.  and    min.    found    by  d\  --  ~  --  7)—°   or 


--  ~  --  7 
vcos  £  cos  til 


tan 


T)=* 


'(7 

\  tan 

Loci:  Absolute  max.,  the  prime-vertical;  absolute  min.,  the  six-hour  circle;  numerical 
max.  and  min.,  the  meridian. 

N   dZ  sin  /  sin  d 

No.  Q.    From  (63),  -TT  =  —  tanZtantf  = ~ 7,  for  error  in  #.     Absolute  mm.. 

'  dd  cos  Zcos  h 

when  Z  =  o,   180°  ;  absolute    max.,   when  Z  =  90°,  270°;  numerical   max.,  when   q  =  90° 
(limited  to  bodies  having  ±  */>  Z),  obvious,  but  may  be  derived  by  algebraic  max.  and  min. 

jf       sin  *      \ 

from  d  \ ~ 7)  =  o. 

\cosZcos  /z/ 

Loci :  Absolute  min.,  the  meridian  ;  absolute  max.,  the  prime-vertical ;  numerical  max., 
the  curve  of  q  =  go0. 

[NOTE. — In  the  Horizon-azimuth,  inspection  of  (48)  and  (50)  shows  that,  at  a  given  place,  d  =  o  is  the  best  con- 
dition for  either  error  in  d  or  error  in  L  ;  and  that  for  a  given  d,  the  lower  the  latitude  the  better.] 

47.    The  determination  of  the  expressions  of  maxiimim  and  minimum  errors  in  the  computed 
azimuth,  due  to  small  errors  in  the  given  parts  of  the  astronomical  triangle. 

In  No.  I,  error  in  7z,  putting  d  \—  — T)  =  o,  it  would  appear  that  the  second  differ- 

&       \tan<7cos^/ 


ential  is  usurping  the  functions  of  the  first  differential  coefficient ;  but  it  must  not  be  so  re- 
garded for  this  problem  of  the  variation  of  the  triangle.  The  equation  that  the  first  differ- 
ential is  derived  from  is  the  general  equation  giving  the  relations  existing  at  all  times  among 
the  parts  of  the  triangle  represented,  the  constants  as  well  as  the  variables,  and  it  is  not  an 
equation  conditioned  on  algebraic  max.  and  min.  existing. 

The  first  differential  coefficient  gives  the  ratio  of  change  of  bearing  to  change  of  altitude 
for  a  given  body  in  a  given  latitude.  But,  this  ratio  having  different  values  for  different 
positions  of  the  body  in  its  diurnal  course,  the  maximum  and  minimum  values  are  sought,  L 
and  d  remaining  constant.  Hence,  regarding  the  first  differential  as  an  isolated  factor,  legiti- 
mately derived,  which  multiplies  the  error  in  altitude  to  produce  the  error  in  the  computed 
azimuth,  we  dispossess  it  of  its  character  as  a  differential,  regard  it  as  a  quantity  susceptible 


AZIMUTH.  29 

of  true  max.  and  min.  values,  and  seek  these  by  putting  its  own  first  differential  equal  to 
zero. 

The  new  variable  q  thus  enters,  but  its  differential  is  eliminated,  knowing  the  relation  ex- 
isting between  it  and  that  of  the  variable  h,  while  L  and  d  are  constants  ;  leaving  dh  a  factor 
in  every  term  of  the  derived  equation,  thus  dividing  out. 

The  equation  with  any  assumed  latitude  is  that  of  a  curve,  on  the  surface  of  the  sphere, 
which  is  intersected  by  parallels  of  declinations  within  certain  limits.  The  intersections  show 
where  the  bodies  having  the  declinations  corresponding  to  the  parallels  must  be  observed  to 
give  algebraic  max.  or  min.  errors  in  the  azimuth.  To  obtain  this  equation,  in  whatsoever 
terms  of  the  astronomical  triangle  it  may  be  expressed,/-  and  d  are  assumed  constant  not  only 
while  deriving  the  partial  differential,  but  while  finding  its  true  max.  and  min.  values ;  hence 
these,  in  this  case,  are  found  by  making  the  second  differential  coefficient — in  the  orignal 
problem  of  finding  simply  the  general  expression  for  error  in  altitude — equal  to  zero. 

Not  so,  however,  in  the  cases  of  error  in  L  and  in  d.  That  we  are  dealing  with  variations 
of  the  astronomical  triangle  is  forcibly  shown  in  these  cases.  Taking  (59)  as  an  instance, 

dZ 

-Tf  =  tan  h  sin  Z,  the  expression  for  error  in  L  is  the  partial   differential    regarding  t  and  d 

constants  ;  that  is,  no  error  in  the  declination  or  in  the  recorded  time  of  observation.  But, 
having  obtained  this  first  differential  coefficient,  it  is  not  the  second  differential  that  is  put 
equal  to  zero  to  obtain  algebraic  max.  and  min.  errors  in  the  azimuth.  We  do,  indeed,  put 
d  (tan  h  sin  Z}  =  o ;  but  this  must  be  solved  regarding  L,  now,  as  constant  as  well  as  d.  For 
the  error  in  L  will  have  its  maximum  effect  if  the  star  is  observed  when  tan  h  sin  zT  attains  its 
maximum  value,  as  the  star  moves  with  its  unchanged  declination  viewed  by  the  observer  in 
his  fixed  latitude — even  though  his  assumed  latitude  is  slightly  in  error.  Similar  considera- 
tions obtain  when  error  in  d  is  concerned. 

It  is  imperative,  then,  that  in  putting  the  differential  of  \hzfactor  of  the  error  equal  to  zero, 

as,  for  example,  (59)  d  (tan  h  sin  Z}  =  o  or  (60)  d  (  —     — ~-\  =  o,  L  and   d  shall  be  regarded 

»      C.OS  /£    j 

constant  in  the  solution,  notwithstanding  the  factor  multiplying  the  error  is  derived  from  a 
variable  L  or  d ;  for  the  change  in  the  aspect  of  the  astronomical  triangle  is  caused  by  all  the 
parts,  other  than  L  and  d,  varying. 

48.  The  preceding  conditions  being  fulfilled,  to  determine  the  relations  existing  among 
the  several  parts  of  the  astronomical  triangle  to  give  a  max.  or  min.  effect  to  the  coefficient 
of  the  error  in  any  given  part  in  the  problem, — when  any  star  is  observed  at  any  place,  the 
latitude  and  declination  both  remaining  fixed,  while  the  star  travels  along  its  path, — it  is  evi- 
dent that  the  resulting  equation  expressed  in  terms  of  L,  d,  and  any  other  one  part,  will  give 
the  value  of  this  part  required  to  fulfil  the  condition  of  most  favorable  or  least  favorable  ob- 
servation. 

' '  ••]  * 

Hence  the  most  convenient  parts  to  form  the  equation    are  L,  d,  and  /,  for  determining 

the  time  to  observe.  L  and  dfbeing  known,  t  can  be  found  from  the  properly  selected  root  of 
the  equation,  taking  care  that  t  is  represented  by  the  same  trigonometric  function  throughout. 
The  conditions,  good  for  the  particular  star,  hold  good  for  any  star  whatsoever ;  hence, 
giving  successive  values  to  d,  while  L  remains  constant,  the  roots  of  the  equation  give  the  hour- 
angles  (/)  corresponding  ;  and  the  curve,  on  the  surface  of  the  sphere,  of  max.  and  min.  values 


30  AZLM-UTH. 

of  the  error  in  the  computed  azimuth  is  the  locus  of  the  intersections  of  the  parallels  of 
declination  with  the  hour-circles  given  by  the  imposed  conditions.  Hence,  in  this  sense,  we 
may  say  that  /  and  d  vary  together  without  denying  the  unvarying  character  of  d  so  far  as 
concerns  the  establishment  of  the  relations  causing  the  coefficient  of  error  to  be  a  maximum 
or  a  minimum.  Giving  different  values  to  the  arbitrary  constant  L,  we  shall  have  on  the  sur- 
face of  the  sphere  a  system  of  curves. 

49.  For  tracing  the  curve,  in  any  particular  latitude,  the  equation  to   the  curve  may  be 
turned  into  terms  of  Z-,  fixed,  and  any  other  two  parts  varying  together.     Thus  the  terms  may 
be  in  trigonometric  functions  of  Z,  d,  and  h,  one  function  adhered  to  for^,  the  next  best  con- 
dition to  L,  d,  and  /  terms  for  determining  at  what  point  to  observe  in  any  given  case. 

But,  for  curve-tracing,  the  most  useful  originally  derived  equation  is  that  consisting  of 
terms  in  Z,  h,  and  Z — the  function,  for  Z  being  the  cosine — for  from  this  equation  we  can  de- 
rive the  equation  to  the  projection  of  the  curve  referred  to  plane  rectangular  co-ordinates. 

The  foregoing  remarks  apply  to  all  the  cases,  in  the  several  methods  of  finding  the  azi- 
muth, remaining  unmemioned.  Briefly,  the  coefficient  of  error  once  legitimately  derived,  its 
max.  or  min.  effect  must  be  sought  by  considering  that  the  latitude  and  declination  do  not 
vary  while  the  aspect  of  the  astronomical  triangle  changes  with  the  progress  of  the  star  along 
its  path — the  diurnal  circle  •=  parallel  of  declination. 

50.  Forms  of  equations  to  the  loci. 

Summary. 

(l)  In  terms  of  Z,  d,  t;  ] 

,  .     ((  For  use  in  finding  position  to  observe  in  any  given  case. 

\2  )  JL^,  ^Z,  rt,   ] 

(3)  "       "       "  L,/i,Z;  referred  to  spherical  polar  co-ordinates. 

(4)  derived  from  (3);  referred  to  spherical  rectangular  co-ordinates. 

(5)  derived  from  (3);  referred  to  plane  polar  co-ordinates. 

(6)  "      (3)  or  (5);  referred  to  plane  rectangular  co-ordinates. 

(l),  (2),  (3),  and  (4)  all  refer  to  the  curve  on  the  surface  of  the  sphere ;  (5)  and  (6)  to  the 
stereographic  projection  of  the  curve. 

(4)  and  (5),  though  interesting,  are  of  little  use  in  this  treatise. 

The  next  step  will  be  to  deduce  the  equations  of  algebraic  max.  and  min.  in  the  several 
cases,  before  discussing  the  curves  they  represent  for  numerical  max.  or  min.  The  equation 
to  equal  numerical  effect  of  error  in  L,  by  both  time-azimuth  and  altitude-azimuth,  will  also  be 
deduced. 


PART  VI. 

EQUATIONS  TO  THE  LOCI  OF  MAXIMUM  AND  MINIMUM 
ERRORS  IN  THE  COMPUTED  AZIMUTH  DUE  TO  ERRORS 
IN  THE  DATA. 


51.  Locus  No.  i.  Altitude-azimuth,  error  in  h  (art.  43).     The  equation  to  that  part  of  the 
locus  of   numerical  mininum   error,  not  obvious,  being  the  locus  of    algebraic  max.  and  min., 

when  ±  d  <  L. 

1st    Equation — in   terms    of   Z,,  d,    and    /.     Of   the    several    equivalent    forms  of     (55) 

any  one  may  be  differentiated  to  give  the  equation,  but  care  must  be  taken  that  a  factor 
essential  to  give  some  part  of  the  entire  curve  shall  not  be  divided  out  during  the  operations 
of  simplifying ;  otherwise,  a  true  branch  may  be  missing  in  the  result :  so,  also,  in  the  remain- 
ing cases  (56),  (57),  etc.  The  labor  of  reduction  varies  with  the  different  forms  used.  In  these 
pages,  the  expression  giving  the  least  labor,  in  each  case,  is  retained,  after  experimenting  with 
the  several  forms  to  ascertain  which  is  preferable,  as  well  as  to  check  .the  result. 

(cos  o\ 
tl  =  °» 
sin    LI 


—  sin  /  sin   q  dq  —  cos   q  cos  /  dt  =  o (77) 


cos  Z  sin  q 
Substituting  from  (65),  dq  = : — - —  at; 

cos  Z  sin8  q  —  cos  q  cos  t  =  o; (78) 

.*.  cosZ(l  —  cos8  q)  —  cos  q  cos  t  =  o,     turn  into  /,  L,  d; 
cos  £(i  —  cos8  C)  —  cos  CcosA  =  o,     turn  into  A,  c,  < 

sin  c  cos  b  —  cos  c  sin  b  cos  A 

By  trig.,  cos  B  —  -     .—  — ; (80) 

sin  a 

sin  b  cos  c  —  cos  b  sin  c  cos  A. 

COS  C = : ' (ol) 

sin  a 

cos  a  =  cos  b  cos  c  -J-  sin  b  sin  c  cos  A; (82) 

sin8  a  —  i  —  cos8  a (83) 


32  AZIMUTH. 

Substituting  (80)  to  (83)  in  (79)  and  reducing, 


cos4  A  -j-  tan  b  cot  c  (cot2  b  —  2}  cos3  A  —  3  cot2  £  cos"  ^ 

-|-  tan  b  cot  c  (2  cosec2  b  -\-  cot2  c)  cos  A  —  cosec2  b  =  o ; 

cos4  *  -I j  (tan2  d—  2}  cos3  *  —  3  tan2  L  cos2  / 

1  tan  d v 

+  —    — >(2  sec2  */-|-  tan2  Z)  cos  t  —  sec2  d  =  o. 
1   tan  */v  ' 


(84) 


52.  2d  Equation  —  in  terms  of  L,  d,  and  /z,  may  be  derived  from  the  foregoing,  but  more 
conveniently  from 


tan  q 


-  --  7]  = 
cos  h) 


(85) 


/.  tan  q  sin  h  dh  —  cos  h  sec2  q  dq  =  o. 


(86) 


From  (64), 


tan  Z  cos 


(87) 


Substituting  and  dividing  out  dk, 


tan  q  sin  ^ 
2 


tan  Z  cos  ^ 


=  o. 


(88) 


/.  sin  q  cos  q  tan  Z  sin  ^  —  I  =  o. 


(89) 


Substituting 


sin  q  = 


sin  Z  cos  L 


cos  d 


we  have 


cos  q  sin2  Z  cos  L  sin 
cos  ^  cos  Z 


—  I  =  o. 


(90) 


cos  ^  (i  —  cos2  Z)  cos  Z  sin  //  —  cos  Z  cos  d  =  o,     turn  into  //,  Z, 
cos  C  (i  —  cos2  .Z?)  sin  £  cos  #  —  cos  B  sin  &  =  o,     turn  into  rt,  £,  & 


"     }      •     (91) 


By  trig., 


cos  /  = 


cos  b  —  cos  c  cos  a 
:  --  .  -  . 
sin  a  sin  ^ 


(92) 


cos  6  = 


cos  c  —  cos  a  cos 
sin  «  sin  b 


(93) 


AZIMUTH.  33 

Substituting  (92),  (93),  and  (83)  in  (91),  and  reducing, 

sin4  h : — 7  (sin2  d-\-  2)  +  3  sin2  L  sin2  h  4-  '— — -7  (2  cos2  d—  sin9  Z-)  sin  h  —  cos1  d  =  o.  (94) 

sintfP  sm*/v 

Note  the  difference  between  (84)  and  (94)  in  the  functions  of  L  and  d. 

53.  3d  Equation — in  terms  of  h,  L,  and  Z. 

From  (79),  cos  B  (i  —  cos2  C)  —  cos  C*  cos  A  =  o,     turn  into  /?,  «,  r. 

Multiply  by  sin2  b. 

sin2  £  cos  B  —  sin2  ^  cos  B  cos2  C  —  sin  b  cos  £7 .  sin  b  cos  ^4.        ...     (95) 

From  trig.,                          sin  b  cos  C  =  sin  «  cos  t:  —  cos  #  sin  c  cos  .# (96) 

sin  b  cos  yi  =  sin  c  cos  tf  —  cos  c  sin  #  cos  B (97) 

sin2  b  —  i  —  cos2  £ (98) 

cos  #  =  cos  c  cos  #  -f-  sin  £  sin  «  cos  /? (99) 

Substituting  (96)  to  (99),  as  required,  in  (95)  and  reducing, 

cos3^-f-  cos  a  sin  a  cot  c  cos2  ^  —  (i  -j-  sin2 «  cot2  c  -f-  cos2  a)  cos  .Z?  -|-  sin  a  cos  a  cot  c  =  o.  (100) 

.  • .  letting  A  represent  tan  Z., 

cos3  ^4"  A  sin  h  cos  ^  cos2  Z  —  (i  -\-  A2  cos2  ^-f-  sin2^)  cos  ^"-j-  A  cos  h  sin  /t  =  o.     .  (101) 
Or,  putting  zenith-distance  in  place  of  the  complement  of  the  altitude, 

cos3  Z  -\-  A  cos  z  sin  #  cos2  Z  —  (i  -|-  A2  sin2  z  +  cos2  z)  cos  Z -\-  A  sin  ^  cos^r  =  o.      .  (102) 

54.  From  (102)  may  readily  be  derived  the  equation  to  the  stereographic  projection  of 
the  curve,  referred  to  plane  rectangular  co-ordinates. 

The  primitive  plane  being  the  horizon,  the  rectangular  axes  X  and  Y  are,  respectively, 
the  projections  of  the  prime-vertical  and  meridian ;  the  origin  being  the  centre  of  the  primi- 
tive circle,  representing  the  zenith ;  the  point  of  sight  situated  at  the  nadir.  To  agree  with 
the  accepted  reckoning  of  /  and  Z  to  the  westward,  x  should  be  reckoned  positive  to  the  west- 
ward and  negative  to  the  eastward.  Hence,  in  north  latitude  x  would  be  positive  to  the 
left,  negative  to  the  right  hand,  and  the  reverse  in  south  latitude ;  the  latter  conforming  to 
the  conventional  reckoning  for  plane  rectangular  co-ordinates. 


34  AZIMUTH. 

So  far  as  tracing  the  curve  is  concerned,  since  there  are  always  branches  symmetrically 
situated  on  each  side  of  Y.  the  meridian,  it  does  not  matter  which  direction  for  x  is  arbitrarily 
chosen  as  positive ;  therefore,  for  convenience,  x  will  be  taken  positive  to  the  right  hand, 
negative  to  the  left,  adhering  to  the  conventional  reckoning,  y  is  reckoned  positive  towards 
the  elevated  pole,  negative  towards  the  depressed  pole. 

55.  To  obtain  the  formulas  of  transformation  of  equation  (102)  to  the  equation  of  the  pro- 
jected curve. 

The  radius  of  the  primitive  circle  being  unity,  z  the  zenith  distance  of  any  point  of  the 
celestial  sphere,  and  r  the  linear  distance,  on  the  projection,  of  this  projected  point  from  the 
centre  of  the  primitive  circle, — we  have,  from  the  principles  of  stereographic  projection, 

r  =  tan%z; (103) 

2  tan  ^z 

and  from  plane  trig.,  sin  z  =  -  ~- (104) 

i  +  tan  \z 

I  —  tan2  4ar 

coz*  =  r+Twn^   •   •   - (105) 


2  tan  \s 

tan  z  =  -        —rr~  ............     (106) 

i  —  tan     z 


Since  the  angle  Z  on  the  sphere  and  in  the  projection  has  the  same  value, 


y 

cos  Z  =  sin  co-Z  —  -  ; 


and,  since  r2  =  x*  -\-y\      ............     (108) 

we  have  for  direct  substitution  in  (101)  or  (102),  to  obtain  the  equation  to  the  projected  curve, 


2r 

~  /IOQ\ 


—  x*  —  y 

,     .     (no) 


(in) 


i  —  r  '      i  —  .*•'  —  y 

cot  z  =  tan  h  =  -       —  =  -  ;  ---  ^_  .  .     .  (\  12) 

* 


2r 


=       '  '  (U3) 


AZIMUTH. 


Let 


A  represent  tan  L  ; 

B  cos  A  whence   |/7~^B2  =  sin  Z  ; 

C         "         sin  A         "        4/r^C2  —  cos  L. 


(114) 


[NOTE.  —  A,  alone,  is  required  in  the  cases  discussed  in  this  treatise  ;  but  based  on  the  principles  contained 
herein,  a  large  number  of  curves  interesting  to  mathematicians  may  be  found,  when  considering  all  the  variations 
of  the  astronomical  triangle,  even  though  the  problems  lack  utility  in  astronomy.  The  writer  has  found,  in  the 
case  of  Time-altitude-latitude  azimuth  (art.  22),  the  necessity  of  employing  either  B  or  C,  in  order  to  introduce 
but  one  arbitrary  constant  (see  Appendix).] 

It  will  be  found  convenient  to  retain  r  until  its  elimination  is  effected  as  far  as  possible 
before  substituting  for  r\  r\  etc.,  x*  +  y,  x*  +  2.x*f  +  y,  etc. 

56.  To  obtain  the  equation  to  No.  i,  for  the  projected  curve,  we  have  from  (101)  or 
(102),  substituting,  as  required,  the  equations  of  art.  55, 


_  _ 

r3  "*     r\i  +  V)a      ~\l   h 


-f 


_ 


2Ar4(i  -  r2)  =  o.  (116) 


Writing  out 


r"  —  x3 


=  o.    (117) 


(118) 


Substituting,  as  required,  (118)  in  (117);  arranging  in  order  of  powers  of  y,  and  changing 
signs  throughout,  to  give  positive  sign  to  the  first  term,  we  have  : 


-f  4A/ 


-f 


—  4A/ 


=  O. 


...   (i  19) 


Rearranging  in  order  of  highest-degree  terms, 


+  4Ay  -  2y  +  8Avy 

-  4Aj4  -  6A;tr2y  - 

+y 


(120) 


This  is  the  equation  to  the  locus  of  algebraic  max.  and  min.,  giving  numerical  min.,  and 
applies  only  when  ±d<^L. 

57.  For  this  case,  the  equation  referred  to  spherical  rectangular  co-ordinates  is  of  too 
high  a  degree  to  be  useful  ;  but  in  some  of  the  subsequent  cases  that  form  will  be  of  interest 


36  AZIMUTH. 

if  not  adding  anything  to  aid  in  the  analysis  of  the  curve.     The  equation  in  this  form  will  be 
introduced  in  Locus  No.  4. 

The  equation  referred  to  plane-polar  co-ordinates  may  be  easily  obtained  by  omitting  to 

y 

substitute  for  cos  Z,  in  the  foregoing  work,  —  —  —  —  =.     But  little  additional  interest  will  be 

yy  -{-  x* 

lent  in  any  case,  and  for  this  locus  the  equation  is  too  complex  to  be  used  advantageously. 

58.  Although  not  a  locus  of  algebraic  max.  and  min.,  the  curve  q  =  90°  is  a  branch  of 
the  locus  of  numerical  min.,  giving  absolute  min.  =  o,  applicable  only  when  d  >  L.  This 
curve,  for  our  purposes,  may  be  called  the  curve  of  elongations,  while  known  to  mathema- 
ticians as  the  spherical  ellipse.* 

The  equations  are  as  follows  : 

tan  L  sin  L  tan  L 

cos  /  =  -  -  j  ;    sin  h  =  —.  —  -,\    cos  Z  =  -  -  7-    ......     (121) 

tan  a  sin  a  tan  h 

Substituting,  from  art.  55,  in  the  last  of  (121),  we  have  for  the  equation  to  the  projection  : 

/  +  2A/—  .?  +  .*>  +  2A**  =  o;    ........     (122) 

or,  y  -\-sfy-\-  2Ay  -\-2.Kx*  —  y  —  o  .........     (123) 


59.     The  remaining  branch,  giving  absolute  max.,  the  meridian,  is  clearly  defined  as  the 
axis  Y.     No  equation  is  needed  ;  but  we  have,  as  a  matter  of  interest, 


cos"  t  —  I  =  o,  cos  t  =  ±  i ;  i 

{cos  (L  —  d, 
or 
—  cos  (L  -f-  d; 
cos*  Z  —  I  =  o,  cos  Z  =  ±  i- 


(124) 


y 

'  *  =  o  .....    ...    (125) 


For  analysis  of  (120)  and  (123)  see  arts.  92,  93. 

60.  Locus  No.  2.    Alt.-az.,  error  in  L  (art.  44). 

There  is  no  branch  of  algebraic  max.  and  min.  For  absolute  max.,  the  meridian,  as  in 
No.  I  (art.  59).  For  absolute  min.,  the  six-hour  circle,  which  is  clearly  defined  by  simple 
construction  according  to  the  principles  of  stereographic  projection  ;  yet,  as  a  matter  of  in- 
terest, we  have  the  equations 

*  For  the  knowledge  th,at  the  curve  of  q  =  90°  is  the  spherical  ellipse,  the  writer  is  indebted  to  Prof.  W.  W. 
Hendrickson,  U.S.N. 


AZIMUTH.  37 

cos  t  =  o  ;         sin  h  =  sin  d  sin  L  ;         cos  ^  =  ^ .  (126) 

tanZ. 


y  -j-  2A.y  -f-  x1  —  i  =  o, 
or,  y  -1-  x*  4-  2Ai/  —  i  =  o. 


Whence,  y  —  —  A  ±  ^Aa  +  i  —  x> (128) 

(129) 


For  analysis  see  art  94. 

6l.  Z-^«J  .Afa.  3.     Alt.-az.,  error  in  d  (art.  44). 

The  meridian  is  a  branch,  giving  absolute  max.  (see  art.  59).     The  six-hour  circle  is  a 
branch  giving  algebraic  max.  and  min.,  always  numerical  min.     This  is  obvious  from  (57), 

dZ  i  /         i         \ 

-jj  =  --  :  —  -  --  r  !  but,  to  derive  the  equation,  we  have  d\-  --  .-    =  o  : 
dd  sin  t  cos  L  \sm  q  cos  h' 

.'.  sin  ^  sin  ^^  —  cos  &  cos  g  dg  =  o  .........     (130) 


By  (64),  dq  =  -. 


.     ^ 
sin  Z  cos 


cos  q  cos  ^ 

sin  0-  sin  h  --  *  —  ~—  =  o 
sin  Z 


sin  ^  sin  Z"  sin  h  —  cos  <?  cos  Z  —  o  ;  ) 

.       *       D  !,        „  >       .......     (132) 

sm  C  sin  B  cos  #  —  cos  C  cos  .5  =  o.   ) 

By  trig.,  the  first  member  of  (132)  =  cos  A-, 

.  *  .  cos  A  =  o  ; 

cos  t  —  o,         t  =  90°  or  270°, 

whence  (126)  to  (129)  recur  (art.  60). 

62.  Locus  No.  4.     Time-azimuth^  error  in  t  (art.  45). 

Curve  of  q  =  90°  is  a  branch  of  absolute  min  (see  art.  58).     Curve  of  algebraic  max.  and 
min.  giving  generally  numerical  max.,  the  meridian  ;  numerical  min.  on  (?)  to  be  found. 

The  equation  in  terms  of  t,  L,  and  d. 

,  /cos  q  sin  Z\ 

From  (58),  d\-  -)  =o. 

\      sin  t      I 

sin  t  sin  g  sin  Z  dg  —  sin  t  cos  $  cos  Z  dZ  -{-  cos  4  sin  Z  cos  t  dt  =  o.    .     .     .     (133) 


38  AZIMUTH. 

cos  Z  sin  q                ,_           cos  q  sin  Z 
From  (65)  and  (58),      dq  =  — ^-f — dt ;         dZ  — ^-^ — (W ; 

.  •.  _  cosZsin2^sin  Z -\-  cos2  ^  sin  ZcosZ-f-  cos  q  sin  Z  cos  /  =  o.       .     .     (134) 
Divide  out  sin  Z  and  substitute  i  —  cos2  q  for  sin2  q, •    •     •     (J35) 

and  we  have  -  cos  Z(i  —2  cos2  q)  -J-  cos  q  cos  t  =  o  ; 

—  cos  .5(1  —  2  cos2  £7)  -j-  cos  (7  cos  A  =  o. 

Now  turn  (136)  into  /,  Z,  d;  A,  c,  b. 

[NOTE. — Though  the  factor  sin  Z,  which  is  divided  out,  gives  the  meridian  when  Z  =  o,  yet  enough  remains  to 
give  it  still.  But,  employing  d  1  —  -^ — I  =  o,  (cos  d  being  constant),  sin  Z  also  divides  out  and  leaves  nothing  to 
represent  the  meridian  (see  art.  51).] 

In  (136),  substituting  (80),  (81),  (82),  and  (83),  and  reducing, 

•      3  •       2    7. 

cos4  A  —  2  tan  b  cot  c  cos3  A  -4-  — .  ,  ,  r-^ —  cos2  A  -J-  2  tan  £  cot  ^  cos  ^ 

sin  #  sin  c 

sin2  £  —  sin2  b  cos3 


sn 


cos2  L cos2  d 

cos4/  —  2  cot  d  tan  £coss*-| 5-7 r-r-  cos2  /  +  2  cot  </ tan  Lcost 

cos  Z,  cos  <z 


cos2  Z  —  cos2  </sin2  Z 

=  O. 


cos2  d  cos 
Factoring  (138), 


(138) 


.(  .   cos2  Z.  —  sin2/,  cos2  df\ 

(cos  /  —  i)  (cos  /  —  2tan  L  cot  <af  cos  /  -\  --       —  -^-j.  --  5—7  —        =  o.     .     .     (139) 
1  \  cos  Z  cos  d        I 

Putting  the  factor  cos2/—  I  =  o,         cos  /  =  ±  I  ;      ...     .....     (14°} 

therefore  the  meridian  is  a  branch  of  the  locus  of  algebraic  max.  and  min.,  giving,  excepting 
in  very  high  latitudes,  always  numerical  max.  (see  art.  59). 
63.  The  other  factor,  solved  as  a  quadratic,  gives 


tan  L        4xsin2  L  —  sin2  d  ,       . 

cos*=-    — 7  ±  -^— -, -: = •     (141) 

tan  a        sin  a  cos  a  cos  L 

It  is  obvious  that  this  branch  exists  only  for  bodies  having  ±  d  <  L  .'.  since -7  >  i, 

the  positive  sign  of  the  radical  is  inadmissible.     This  branch  is  that  of  numerical  minimum. 
64.  The  equation  in  terms  of  h,  L,  and  d. 


AZIMUTH.  39 

sin  h  —  sin  L  sin  d 

In  (139),  substitute  cos/= j —       , (142) 

v  JW  cos  L  cos  d 

The  first  factor  becomes     (sin  h  —  sin  L  sin  d}*  —  cos"  L  cos"  d=  o; (H3) 

.  • .  sin  h  —  sin  Lsind  =  ±  cosL  cosd; (144) 

(        sin  Z-sin  d-\-  cos  L  cos  </=  cos  (Z,  —  <aO,       ) 

.•.sinA=H         .         .  '       \     .    .    .    .    (145) 

(  or  sin  L  sin  d  —  cos  Z  cos  d  =  —  cos  (Z  -j-  ^)'  ) 

From  (145),  the  meridian  is  a  branch  of  the  locus  of  algebraic  max.  and  min. 
The  second  factor  of  (139)  becomes,  by  substituting  (142), 

sin  dfsin2  h  —  2  sin  L  sin  h  -(-  sin  d=.  o; (146) 


.-.  solving  as  a  quadratic,  sin  h  =  sm  L  ±  ]  ^^ *—. (147) 

The  radical  shows  that  </>Z  gives  an  imaginary  result ;  therefore  this  branch  exists  only 
for  </<Z;  .'.  since  -  — j  >  I,  the  positive  sign  of  the  radical  is  inadmissible.     This  branch 

Sill  & 

gives  numerical  min.     The  whole  curve  is  given  by 

[(sin  h  —  sin  Z  sin  d}*  —  cos2  L  cos2  d~\  [sin2  h  sin  d  —  2  sin  Z  sin  h  -f-  sin  d~\  ;     (148) 


or  sin*  h  —  ^-^ — .     \,  S1"      ^  sin3  h  4-  (5  sin2  Z  +  sin2  //)  sin2  £ 

sin  </ 

,    2  sin  L(i  —  sin2  Z  —  2  sins^)    , 

— s — j sm  h  —  (i  —  sin  L  —  sin8  ^)  =  o. 

sin  d  v  y 


^(149) 


65.  The  equation  in  terms  of  Z,  L,  h. 

Taking  (136),        cos  B  —  2  cos  B  cos2  C  —  cos  C  cos  A,     turn  into  C,  B,  a. 
Multiplying  by  sin"  b, 

sin2  b  cos  B  —  2  cos  .Z?  sin2  b  cos2  C  —  sin  b  cos  C  .  sin  b  cos  ^4.     ...     (150) 

Substituting  (96),  (97),  (98),  (99),  in  (150)  and  reducing, 


40  AZIMUTH. 

cos  a  sin  a  cos  c  cos  #  sin  a  cos 

COS    B  —  —  ,     N  COS    .Z?  —  COo  #  4-  -:  -  T  —  -,  --  —  { 

sin  4  1  +  cos2  a)  '    sin  <:([  +  cos*  #) 

Letting  A  =  tan  Z,, 

A  sin  h  cos  /;  _   .    A  sin  h  cos  h 


sn 


(151) 


A  sin  h  cos  k 
Factoring,  (cos2  Z—  i)  ^cos  Z  --  ~ 


The  first  factor  defines  the  meridian  ;  for  the  remaining  branch  we  have 

A  sin  h  cos  h 


66.  The  equation  to  the  stereographic  projection  of  the  branch  given  by  (153). 
Substituting  in  (153)  as  needed,  from  art.  55,  and  reducing, 


o;      .          .     (154) 
or,  rearranging, 

4-  A/-  A*9  +.7  =  0.      .    .    .    (155) 


For  analysis  see  art.  95. 

67.  The  equation  referred  to  spherical  rectangular  co-ordinates. 

The  axes  X  and  Y  being,  respectively,  the  prime-vertical  and  the  meridian,  the  origin  at 
the  zenith,  x,  in  arc,  may  be  reckoned  positive  to  the  westward  around  to  360°,  or  positive 
west  to  180°  and  negative  east  to  180°  ;  y,  positive  towards  the  elevated  pole  up  to  90°,  the 
point  in  the  horizon  which  is  the  pole  to  the  prime-vertical  ;  negative  to  90°  towards  the 
other  pole. 

sin  y       sin  y 
We  have  then,  cos  Z  =  sin  co-Z  =  ^—  —  =  --  -f;        .......     (156) 

sin  2       cos  h 

cos  z  =  sin  h  =  cos  x  cos^/  ...........     (1S7) 

Substituting  (156  and  (157)  in  (153), 

sinjj/  _  A  cos  x  cos  y  cos  h 
cos  h  ~      cosa  x  cos2  y  +  I 

.  *.  sin  y  cos*  x  cos*^/  +  sin^/  =  A  cos  x  cos_7  cos*  h  ......     (J59) 


AZIMUTH.  4I 

Then,  since  cos"  h  —  I  —  sin"  h  =  I  —  cos2  x  cos2j/;     .......     (160) 

sin  y  cos"  ^r  cosa  ^  -(-  sin  ^  =  A  cos  x  cos  _y  —  A  cos3  x  cos8  ^  ;     .     .     .     (161) 

.  *  .  A  cos3  y  cos3  x  -j-  sin  y  cos"  j/  cos2  ^  —  A  cos  y  cos  x  -f-  sin  _y  =  o  ;    .     .     (162) 

i 
sinj  I  sinj 

COS    X-\--j:  -  -  —  COS    X  --  —  COS;r4--T  --  —  =  O  ......      (1V$) 

1   Acosj/  cos  y  A  cos  y 

For  the  branch  q  =  90°,  see  art.  58. 

68.  Locus  No.  5.     Time-azimuth,  error  in  L  (art.  45). 

The  meridian  and  the  horizon  define  the  absolute  min.     For  numerical  max.,  the  curve 
of  algebraic  max.  and  min.  as  follows  : 

The  equation  in  terms  of  /,  Z,  d. 

From  (59),  */  (tan  h  sin  Z)  =  o  ; 

.  •  .  tan  h  cos  ZdZ  -\-  sin  Z  sec2  h  dh  =  o  ........ 


By  (55), 


-  -  -       7 
tan      cos  h 


.  •  .  sin  h  +  tan  ^  tan  q  =  o,    turn  into  t,  L,d. 


cos  #  -f-  tan  ^  tan  £T  =  o,    turn  into  A 


,.\ 
,  c,  &.) 


Multiply  (165)  by  sin2  A  cot  A  cot  B,  and  in  the  resulting  second  term  put  i  —  cos2  A  for 
sin2  A,  and  we  have 

cos  a  cot  B  sin  A  cot  £7  sin  ^  -J-  i  —  cos8  A  —Q  ......  (J66) 

By  trig.,                              cot  B  sin  ^4  =  sin  c  cot  #  —  cos  c  cos  /2  ;       .......  (l^7) 

cot  C  sin  y2  —  sin  b  cot  ^  —  cos  b  cos  ^  ;       .......  068) 

cos  #  —  cos  b  cos  ^  -f-  sin  b  sin  £  cos  A  .........  (l&9) 

In  (166),  substituting  (167),  (168),  (169),  and  reducing,  we  have 

s          3  sin*  Z  sin8  </  —  sin2  L  —  sin2  d  —  I 
cos*  *  +  -  -  .          .  —,  --  -  --  cos2  t 

sin  fz  sin  Z,  cos  d  cos  Z 

-(170) 

+  •        <2          r         *        '2        T  \// 

sin   «  sin   Z. 

-j-  1  1  —  tan-  L.  —  tan-  a  )  cos  r  -+-  -.  —  -r-.—^—    —  :  -  F  =  O. 

sin  <at  sin  L  cos  */  cos  L 


42  AZIMUTH. 

69.  The  equation  in  terms  of  h,  L,  d. 

From  (165),  sin  h  cos  Z  cos  q  -\-  sin  Z  sin  q  =  o ; 

sin  a  cos  d 


substituting,  smZ= 


cos 


sin  h  cos  Z  cos  q  cos  Z  -f-  sin2  q  cos  d  —  o;   turn  into  h,  L,  d\ 
cos  #  cos  ^  cos  £7  sin  £  4~  sin  #  —  sin  b  cos8  £7  =  o  ;   turn  into 

In  (172)  substituting  (92),  (93),  and  (83)  and  reducing, 


j 
a,  c,  b.     } 


i  4-  cos5  c  -4-  cos"  b  .    i  —  ccsa  ^  —  cos3  b                         . 
cos  #  --        —  T—            -  cos8  #  4-  3  cos  rt  H  ---  7  --  =  O  ;      .     .     (173) 

cos  b  cos  c  cos  <?  cos  c 

I  +  sin*  Z  4-  sin"  d   .  I  —  sin"  L  —  sin"  </ 

sin3  ^  --    —  :  —  j-.--r  -  sin2  //  +  3  sin  h  -\  --  =  —  -r-.  —  j  -  =  o.       .     .     (174) 

sin  d  sin  L  sin  d  sin  L 

70.  The  equation  in  terms  of  Z,  L,  h. 
Multiplying  (172)  by  sin  b, 


cos  a  sin  c  cos  B  sin  b  cos  67  -f-  sin"  b  —  sin"  b  cos4  (7  =  o  .....     (17$) 
Substituting  (96),  (98),  (99),  and  reducing, 

cot  c  sin  a  cos  a  I 

cosa  B  --  —  ^—   -  cos  B  --  :  ---  —  =  o;     .....     (176) 

i  4~  cos  a  i  -f-  cos*  a 

tanZ,  cos^sin/j  i 

COS    Z  --  ;  --  r-f-,  -  COS  Z  --  j  -  .—  ry  =  O  ......       (I77) 

i  4-  sm  h  i  4~  sm  ^ 

71.  The  equation  to  the  stereographic  projection  of  the  curve. 

In  (177)  substitute  from  art.  55,  as  needed,  reducing  and  arranging  in  order  of  powers  of  y, 


-  x 


) 

-  2x"  -  a?  =  o;  f  '  V  7  ; 


or,  rearranging  in  order  of  terms  of  highest  degree, 


_  2J.-4  2Av3 "  A  "*"    '     "*         ""    '  *         ' -'•' 


For  analysis  see  art.  97. 

72.  Locus  No.  6.     Time-azimuth,  error  in  d  (art.  45). 


AZIMUTH.  43 

Curve  of  absolute  min.,  the  meridian ;  curve  of  numerical  max.  and  min.,  from  algebraic 
max.  and  min.,  as  follows  : 

The  equation  in  terms  of  t,  L,  d. 

From  (60),  d( A  =  o,        cos  h  cos  q  dq-\-  sin  q  sin  h  dh  =  o.      ....    (180) 

By  (64),  dq  = ^ 7  dh ; 

tan  Z  cos  h 

COS    Q 

' 


cos  q  cot  Z-\-  sin  q  sin  h  =  O 

Dividing  by  sin  q, 

cot  q  cot  Z-\-  sin  h  =  o;    turn  into  t,  Z,  d;  ) 
cot  C  cot  ^  +  cos  a  =  o ;    turn  into  A,  c,  b.  f ' 

Multiply  by  sin2  A  =  i  —  cos2  A, 

sin  A  cot  C1  sin  ^4  cot  B  -\-  (i  —  cos2  y2)  cos  #  =  o.     .     .     .     .     .     (184) 

Substituting  (167),  (168),  (169),  and  reducing, 

cos3  A  -f-  (cot2  c  +  cot3  b  —  i)  cos  A  —  2  cot  b  cot  c  =  o;     .     .     .     .     (185) 
cos3  t  -\-  (tan2  L  -f-  tan2  d  —  i)  cos  /  —  2  tan  */  tan  L  =  o 086) 

73.  The  equation  in  terms  of  h,  L,  d. 

e  ,    L.     ,.  sin  //  —  sin  <af  sin  L  .     ,  _ 

Substituting  cos  /  = 3 = m  (186), 

cos  d  cos  Z 

and  we  have 

sin8  h  —  3  sin  Z  sin  */  sin2  h  -f-  (2  sin8  Z  -|-  2  sin2  </  —  i)  sin  h  —  sin  <a?  sin  Z  =  o.     (187) 

74.  The  equation  in  terms  of  Z,  Z,  h. 

From  (60),  d\^ — )  =  o; 

\  sm  /  / 

2  sin  /  sin  Z  cos  ZdZ  —  sin2  Z  cos  t  dt  •=  Q 088) 


44  AZIMUTH. 


cos  a  sin 
By  (62),  rfZ=  --     - 


.  •  .  2  cos  q  cos  Z  -)-  cos  t  =  o  ;    turn  into  Z,L,h;\ 
2  cos  6"  cos  ^  -|-  cos  ^4  =  o;    turn  into  .5,  r,  #.   f 

Multiplying  by  sin  b,  2  sin  £  cos  C  cos  -5  4~  sin  b  cos  ^  =  o  ........     09°) 

Substituting  (96)  and  (97)  in  (190)  and  reducing, 

cos2  B  —  %  tan  a  cot  c  cos  .#  —  £  =  o.      .     ,     .....     (191} 


-  ,   .       .,  j     ..  „,      tan  Z  ±  ^tan2  L  -4-  8  tan4  /?  /T^^\ 

Solving  the  quadratic,  cos  Z  =  -  ="  --  —  --  ........     (193) 

4  tan  h 

75.  The  equation  to  the  stereographic  projection  of  curve  No.  6. 
In  (192)  substituting  as  needed  from  art.  55  and  reducing, 


/  +  2A/  -  /  +  2A*>  -f  x*  -  x*  =  o  ;     .......     (194) 

or,  /  —  *'  +  2A/  +  2A^r>  —  /  +  x*  =  o  ......     0     .     (1940) 


For  analysis,  see  art.  98. 

76.     Locus  No.  7.     Time-alt.  -azimuth,  error  in  h  (art.  46). 

The  meridian  and  the  horizon  are  branches  giving  absolute  min.  ;  the  prime-vertical  giv- 
ing absolute  max.  The  branch  of  numerical  max.  given  by  algebraic  max.  and  min.  as 
follows  : 

From  (61),  <a?(tan  Ztan  h)  =  o  ; 

tan  Z  sz?  h  dh  -\-  tan  hszc'ZdZ=  o  .........     (19$) 


By  (5  5),  <tZ=—       -—jdh, 

tan  q  cos  h 


sin  Z  sin  /z  cos  ^ 

~~ 


*  cos  Z  cos"  h  '   cos2  h  cos2  Zsin  q 

Substituting,  sin  Z= T — > 

cos  L 

sin2? cos  dcosZ-\-  sin  A  cos  g cos  L  =  o (!97) 


Since 


AZIMUTH. 
sin*  q  =.  i  —  cos*^, 


45 


cos  */cos  Z  —  cos  d  cos  Z  cos*  q  4-  sin  h  cos  ^  cos  Z  =  o  ;     turn  into  *,  Z,  </; 
sin  £  cos  5  —  sin  ^  cos  Z?  cos*  £T  -)-  cos  #  cos  f  sin  c  =  o ;     turn  into  A ,  c,  b ; 


Multiplying  by  sin3  a, 


sin*  a  sin  b  cos  Z?  sin  a  —  sin  £  sin  a  cos  /?  sin"  #  cos*  C-\-  sin*  #  cos  a  sin  ^  cos  C  sin  a  =  o.     (199 


By  trig., 


cos  B  sin  a  =  sin  c  cos  b  —  cos  c  sin  b  cos  ^4 
cos  C  sin  #  =  sin  b  cos  £  —  cos  b  sin  ^  cos  ^4 

sin2  a  =  i  —  cos* «  ; 

cos  #  =  cos  ^  cos  c  -j-  sin  #  sin  c  cos  ^4. 


(200) 


Substituting  (200)  in  (199)  and  reducing, 


cos4  /  4-  (4  tan  d  tan  L  -4-  -.  —  j  --  J  cos3  1 
v*  '   sin  tf  cos  rf/ 

4-   |  3  tan*  Z  (tan*  ^  —  i)  ---  5-^  —  5-^  [ 
I  J  '       cos  d  cos  Z  ) 

/  tan  Z  tan  */\ 

—  1  4  tan  d  tan  Z  -\  ----  =—  j  —  j  cos 
\ 


cos 


cos*  ^  cos4  Z 


cos  rf 
—  tan*  d  tan4  Z  =  o. 


.     .     .     (201) 


[NOTE. — The  writer  has  done  a  great  amount  of  "  dead  work"  in  attempting  to  simplify  this  equation,  as  well  as 
the  other  complex  equations  in  this  treatise.  In  the  end,  of  the  many  various  forms  obtained  in  any  instance,  that 
form  has  been  retained  that  seemed,  taken  all  in  all,  to  be  the  simplest.  The  remaining  forms  found  are  not  given 
here,  though  perhaps  some  of  them  would  appear  to  the  reader  preferable.  The  writer  would  be  glad  to  see  the  vari- 
ous equations  simplified  by  any  one  that  will  attempt  the  work. 

It  may  be  remarked  that  in  performing  the  operations  indicated  in  this  treatise — and  believed  to  be  indicated 
fully  as  to  steps  taken — to  obtain  the  equations  of  high  degree  in  trigonometric  functions,  there  is  a  vast  amount  of 
trigonometric  gymnastics  required  in  order  to  present  the  equations  not  more  complex  than  shown.  After  perform- 
ing the  operations  step  by  step,  and  collecting  the  terms  (as  they  then  stand)  in  the  order  of  highest  degree  terms  of  the 
function  of  that  particular  part  of  the  triangle  that  is  given  but  one  function,  the  coefficients  of  the  different  powers 
of  this  function  are  often  very  complex.  Putting  the  entire  computation  into  these  pages  would  encumber  the  work.] 

In  symplifying  the  coefficients  just  mentioned,  the  formula  found  most  useful  is  sin*  x 
-f-  cos*  x  =  i  ;  and  from  it  sin4  x  =  sin*  x  (i  —  cos*;tr)  and  cos4  x  —  cos*;r  (i  —  sin*  x)  are  often 
needed. 

77.     The  equation  in  terms  of  h,  Z,  d. 

In  (198),  substituting  (92),  (93)  and  (83),  and  reducing, 

.   4.  sin3Z        .  2-|-sin*Z  .  3  sin*/ sin  Z   .         ,   cos*  d  —  sin"  Z 

sin  h  +  j-y  sin3  h —^rj—  sin*  h  -f ^-f sin  h  -4-  -        —-? =  o.     (202) 

sin  at  cos  Z  cos  Z  cos  Z  coe  Z 


4  6  AZIMUTH. 

78.  The  equation  in  terms  of  Z,  L,  h. 
Multiplying  (198)  by  sin  b, 

sin2  b  cos  B  —  sin2  £  cos"  CcosB-\-  cos  a  sin  csin  b  cos  C=  O (203) 

In  (203),  substituting  (96),  (98),  (99),  and  reducing, 

cos3  B  —  sina  a  cos  B  —  cos  a  sin  a  cot  c    =  o (204 

cos3Z—  cos*  A  cos2T—  sin  ^  cos /z  tan  L  —  o (205) 

79.  The  equation  to  the  projection. 

In  (205),  substituting  from  art.  55  as  needed,  and  we  have 

,  ,  j  (206) 

~r  y  ~T  oA^-y  —  4A#y  —  43 :  y  -f-  2A;r  —  2A;r  =  O;  ) 

or,  y1  -\-  2x*y*  -f-  x*y*  -\-  2 Ay*  -f-  6A,ry  -|-  6A.ry  -f-  2 A^r8  | 

[For  analysis,  see  art.  99.] 

80.  Locus  No.  8.     Time-alt. -azimuth,  error  in  t  (art.  46). 

The  prime-vertical  is  a  branch  for  absolute  max.  The  six-hour  circle  is  a  branch  for  ab- 
solute min.  The  branch  of  numerical  max.  and  min.  given  by  algebraic  max.  and  min.,  as 
follows  : 

The  equation  in  terms  of  t,  Z,  and  d. 

From  (62),  */(tan  Z  cot  /)  =  o  ; 

tan  t  sec  ^  d/^,  —  tan  £  sec  /  dt  ^^  o.  ••••••••     (200) 

By  (58), 


hence,  (208)  reduces  to  cos  q  cos  t  -4-  cos  Z=  o  ;  ) 

cos  C  cos  A  -\-  cos  B  =  o  I  )      ......... 


which  from  trig,  give         sin  A  sin  C  cos  £  =  o  ; 

si 

sin  /  cos  L 


sin  /  sin  q  sin  </  =  o  ;  turn  into 

c   ,    ,  ..          .  sin  /  cos 

Substitute  sin  q  =.  —     —  T     , 
2  cos  h 


j 
/,  L,  d.  ) 


AZIMUTH.  47 

and  we  have  sin"  /  cos  L  sin  d  =  o  ;  } 

(cos*  t  —  i)  cos  L  sin  d  =  o.  f 
i 
Hence  with  any  latitude  and  declination, 

cos2  /  —  i  =  o,  cos  t  =  ±  i,  t  =  o°,  180°  ; 

giving  the  meridian. 

8l.  The  equation  in  terms  of  k,  L,  d. 

sin  h  —  sin  L  sin  d 

In  (2ii),  substitute  cos  /  =  -  ?  -  -j  -  ;     .     .          .     . 

cos  L  cos  d 


or  the  same  result  as  follows  : 

cos  a  —  cos  b  cos  c 
By  trig.,  cos^  =  -  —  -j—.  --  ..........     (213) 

sin  b  sin  c  x     J/ 

In  (209),  substitute  (213),  (92),  (93),  and,  reducing,  we  have 

cos  6  (cos3  a  —  2  cose  cos  6  cos  a-\-  cos*  c  cosa£  —  sin"  £  sinV)  =  o;  .     .     .     (214) 
.*.  (cos  a  —  cos  b  cos  c)*  —  sin*  b  sin"  c  =  o  ; 


(215) 
(sin  h  —  sin  d  sin  Z,)a  —  cos2  dcos*  L=  o  ;  J 

The  first  factor  in  (214),  sin  d,  as  a  constant,  divides  out;  the  second  factor,  put  equal  to 
zero  and  solved,  gives,  from  (215), 

sin  h  —  sin  d  s\n  L=  ±  cos  d  cos  Z;     .......     (2J6) 

t 

.'.  sin  /*=  1 
( 


sin  L  sin  d-\-  cos  L  cos  d  —  cos  (L  —  d\  ==  cos  (<a? 

,     .     ,    .     ,  r          ,  ,r    ,     ,x 

and  sin  Lsma  —  cos  Z,  cos  d——  cos  (Z.  -j-  d  ), 


(217)  giving  the  meridian. 

82.  The  equation  in  terms  of  Z,  L,  h. 

Multiplying  (209)  by  sin5  b, 

sin  b  cos  A  sin  b  cos  £"-[-  sin*  b  cos  £  =  o  .....     •     •     (218) 
Substituting  (96),  (97),  (98)  and  (99),  in  (218),  and  reducing, 

cos8  B  -\-  cot  c  cot  a  cos*  B  —  cos  B  —  cot  c  cot  0=0;) 
cos3  Z"--  tan  L  tan  ^  cos2  Z  —  cos  2T  —  tan  L  tan  ^  =  o.  ) 


48  AZIMUTH, 

Factoring  (219),  (cos*  Z  —  i)  (cos  Z -\-  tan  L  tan  K)  =o.    ,^.    i     .     .     .     .     .     (220) 

The  first  factor  gives  the  meridian,  cos  Z =  ±  i,Z  =  o,  180°.  The  second  factor  gives  the 
equator  from 

cos  Z  cos  L  cos  h  -f-  sin  L  sin  h  =  o  ;  ) 

f       ....    (220 
cos  B  sin  £  sin  a  -f-  cos  £  cos  a  =  o  —  cos  b  (by  trig.)  ;  ) 

.•.  sin  d  =  o,  giving  equator.  But,  as  will  be  shown,  this  is  not  a  branch  giving  max.  and 
min.,  but  giving  a  constant  value  to  dZ;  and  the  equator  divides  the  meridian  into  parts 
such  that  the  true  max.  and  min.  at  transits  of  stars  are  for  -f-  d  numerical  max.  at  upper 
culmination,  numerical  min.  at  lower  culmination,  and  vice  versa  for  —  d,  provided  we  do  not 
yet  consider  the  intervening  absolute  max.  and  min. 

83.  Although  the  equator  is  clearly  defined  and  easily  constructed  in  the  projection,  yet 
we  may  find  the  equation  to  its  projection.    From  (220),  cos  Z-\-  tan  L  tan  h  =  o,  and  by  art.  55, 

y       A(i  —  r3) 

=  o ; (222) 

y  27* 

2y  -f  A  -  Ax*  -  A/  =  o ; 
.-.     /  +  3*  —  -^y  —  i  =  o (223) 

A  star  whose  d=  o  is  as  favorably  situated  at  one  point  as  at  any  other  in  its  diurnal 
course.     This  may  be  shown  directly  from  (62), 

dZ       cos  t  cos  d  , 

-i7  = ~ ,- (224) 

dt       cos  Z  cos  n 

By  trig.,  sin  a  cos  B  =  sin  c  cos  b  —  cos  c  sin  b  cos  A  ;   ) 

I  ^  ^  C  i 

.*.  cos  h  cos  Z  =  cos  L  s\n  d  —  sin  7.  cos  */  cos  t ',  } 

when  d  =  o,  cos  //  cos  Z  =  —  sin  L  cos  / (226) 

Substituting  (226)  in  (224),  cos  d  being  unity, 

dZ  i 


dt  sin  L  ' 


(227) 


a  constant  quantity  in  a  given  latitude.     For  analysis,  see  art.  100. 

84.  Locus  No.  g.     Time  alt.  azimuth,  error  in  d  (art.  46). 

The  meridian  is  a  branch  of  absolute  min.     The  prime-vertical  is  a  branch  of  absolute 
max.     The  branch  of  algebraic  max.  and  min.  exists  only  for  ±  d  >  L,  and  it  is  obviously 


AZIMUTH. 


49 


dZ 

the  curve  q  =  90°,  as  shown  in  (63),  -77  —  —  tan  Ztan  d.     Though  obvious,  we  may  find  this 

branch  analytically,  as  one  of  algebraic  max.  and  min.,  as  follows  :  choosing  from  (63),  we  have 


.  ,  sin  q 

d\ %,)  =  o; 

cos 


cos  Z  cos  q  dq  -f-  sin  q  sin  Z  dZ  =  o. 


(228) 


By  (66), 


dZ  = 


cos  q  cos  d 
--  =  --  =*&  ; 
cos  Z  cos  Z,  z 


,    sin  q  sin  .Z  cos  q  cos  ^ 

.  • .  cos  Z  cos  #  -j ^ -f =  o. 

cos  Zcos  L 


(229) 


Substitute 


_  sin  Z  cos  L 

sin  o  —  : 

cos  d 


and  clear  of  fractions,  when  we  have 


cos"  Z  cos  q  -(-  sin"  Z  cos  q  —  o ; 

/cos*  Z  +  sin4  Z\ 

(  _  j          J  cos  ?  =  o. 


(230) 


cos  ^  =  o, 


tan 


cos  /  —  - 


sin   /  = 


tan  d' 

sin  L 
sin  d  ' 

tan  L 
tan  h' 


.  J90C 


(230 


For  the  equation  to  the  projection  of  q  =  90,  see  art.  58,  Locus  No.  I,  where  this  curve  ex- 
ists for  an  absolue  min.  Eq.  (123),  y*  -j-  x*y  -f-  2 Ay  -f-  2A.T2  —  y  =  o  (see  art.  101). 

85.  Locus  No.  10.  Time-azimuth  and  altitude- azimuth  giving  equal  numerical  values 
to  the  error  in  the  computed  azimuth,  arising  from  a  small  error  in  the  latitude ;  whence  the 
limits  within  which  the  one  method  is  more  favorable  than  the  other,  so  far  as  the  error  in  lati- 
tude is  concerned. 

By  inspection  of  (56)  and  (59),  taking  for  investigation  the  single  case  of  a  rising-and- 
setting  body  having  -\- d>  L,  we  see  that  in  the  method  of  altitude-azimuth  the  error  when 
the  body  is  in  the  horizon  at  rising,  t  lying  between  180°  and  270°,  has  a  positive  finite  value, 
decreasing  to  zero  when  t  becomes  equal  to  90°  as  the  star  travels  in  its  diurnal  path ;  by  the 


So  AZIMUTH. 

time-azimuth,  when  the  star  is  in  the  horizon  the  value  of  the  error  is  zero,  and  then  it  in- 
creases  negatively  to  a  negative  finite  value  when  t  =  90°  ;  Z  throughout  this  time  remaining 
between  360°  and  270°. 

.Therefore,  (l)  for  some  point  of  observation  (e')  in  the  star's  path,  between  the  horizon 
and  the  six-hour  circle,  the  values  of  the  error  in  the  azimuths,  computed  with  both  methods 
separately,  will  be  numerically  equal,  with  contrary  signs ;  (2)  it  is  obvious  also  that  before 
the  star  reaches  the  meridian,  after  passing  (*'),  some  point  of  observation  (/"')  will  give 
identical  values  to  the  errors,  that  is,  numerically  equal,  having  the  same  sign. 

The  time-azimuth  then  will  be  preferable  from  the  time  of  rising  of  the  star  until  e  is 
reached  ;  thenceforward,  passing  through  t  =  90°  and  up  to  (/'),  the  altitude-azimuth  the 
better  method;  thereafter  to  the  meridian  we  should  employ  the  time-azimuth.  We  may, 
in  a  similar  way,  follow  the  star  west  of  the  meridian. 

Our  object,  now,  will  be  to  find  the  equations  to  the  curves  of  equal  errors  with  opposite 
signs,  and  with  like  signs ;  whence,  constructing  the  curves,  the  limits  within  which  either 
method  is  preferable  will  be  graphically  given. 

86  (a).  First. — Identical  errors — signs  alike. 

Putting  (56)  and  (59)  equal  to  each  other, 


—f  =  tan  h  sin  Z\ 
tan  /  cos  L 


cot  /  —  tan  h  sin  Z  cos  L    =  o  ;  ) 

I 
cot  A  —  cot  a  sin  B  sin  c    —  o.  ) 


Multiplying  by  sin  B,  sin  B  cot  A  —  cot  a  sin2  B  sin  c  =  o  ........     (234) 

By  trig.,  sin  B  cot  A  =  sin  c  cot  a  —  cos  c  cos  B  ;    } 

sin2  B  =  i  -  cos1  B.  \  ...... 


Substituting  (235)  in  (234)  and  reducing, 

cos"  B  cot  a  sin  c  —  cos  B  cos  c    =  o  ;   ) 

cos2  Z  i  an  h  cos  L  —  cos  Z  sin  L  =  o;    >  .......     (236) 

cos  Z(cos  Z  tan  h  cos  L  —  sin  L  =  o.    ) 

The  first  factor  of  (236)  gives  cos  Z  =  o  ;       ............     (237) 

.'.  Z=go°,  270°,  and  the  prime-  vertical  is  a  branch  of  the  locus. 
Putting  the  second  factor  equal  to  zero,  we  have 

„      tan  L 


AZIMUTH.  51 

which  value  can  exist  only  when  q  =  90°,  270°  ;  hence  the  curve  of  elongations,  q  =  90°,  is  a 
branch  of  the  locus. 

Hence  the  prime  vertical  is  a  branch  for  all  bodies  having  ±  d  <  L,  and  we  have  for  con- 
ditions, 

tan  d  sin  d 

cos  t  =  -  —f,          sin  h  =  —  —  j  ;    .........     (239) 

tan  L  sin  L 

and  the  curves  of  elongations  are  branches  for  ±  d  >  £,  giving 

tan  L  sin  L 

cos  t  =  -  -  ->,         sin  It  --  —  —  -g  ..........     (240) 

tan  a  sin  d 

From  (238)  we  have,  as  in  (123),  the  equation  to  the  projection  of  the  curve  of  elongations, 

/  +  *?y  +  2  A/  -f  2A*8  —  y  —  o  ; 

and  from  (237)  for  the  prime-vertical  y=  o  equation  to  the  projection. 

86  (&).  The  branches  found  in  art.  86  (a)  may  be  determined  with  less  labor  by  selecting 
from  (56)  and  (59)  as  follows  : 

cos  t  sin  h  sin  Z  sin  q 

- 


r 
sin  t  cos  L  sin  t  cos  L 

but  both  methods  are  retained  as  checks  upon  each  other. 

From  (232)^  we  have  cos  t  —  sin  h  sin  Z  sin  q  =  o  ; 

cos  A  —  cos  a  sin  B  sin  C  =  o. 

By  trig.,  -  cos  B  cos  C  —  (233)^  ; 

.  •  .  cos  B  cos  C  =  o  ;  ) 

_  .....     .....     (236)* 

cos  Z  cos  q  —  o.   ) 

.  *  .  cos  Z  =  o  (gives  the  prime-vertical,  and) 
cos  q  =  o  (gives  the  curve  of  elongations.) 

87.  Second.  —  Equal  errors,  numerical,  opposite  signs. 

Changing  the  sign  of  one  member  in  (232)^,  and  cancelling  sin  t  cos  Z,  we  have 

cos  /  ==  —  sin  h  sin  Z  sin  q  ;        j 
cos  A  -f-  cos  a  sin  B  sin  C  =  O.  ) 


52  AZIMUTH. 

For  the  equation  in  terms  of  t,  L,  d\  A,  c,  b. 

sin  A  sin  b  sin  A  sin  c 

By  trig..  sin  B  =  -  :  --         and         sin  C  =  -  :  -  : 

sin  a  sin  a 

cos  a  sin  b  sin  c  sin*  A 
substituting  in  (241),  cos  A  -)  ---  —  --  =  o. 

ol  1  1      c£ 

• 

Substituting  sin*  a  —  I  —  cos"  a,        and        sin8  ^4  =  I  —  cos*  A, 

and  clearing,  we  have 

cos  A  —  cos"  a  cos  /2  -f-  sin  £  sin  t  cos  #  —  sin  £  sin  c  cos  #  cosa  A  =  o.  .     .     (242) 
Since  cos  #  =  cos  b  cos  r  -j-  sin  b  sin  £  cos  A, 

we  have        cos  ^4  —  cos8  b  cos8  £  cos  A  —  2  cos  #  cos  *:  sin  £  sin  c  cos8  y2  } 

—  sin8  b  sin8  £  cos3  A  -f-  sin  £  sin  ^  cos  £  cos  £  -j-  sin8  £  sin8  c  cos  ,/2      >•  (243) 

—  sin  b  sin  £  cos  b  cos  £  cos8  A  —  sin8  £  sin8  c  cos3  ^4  =  o.  ) 

Arranging  in  order  of  powers  of  cos  A,  collecting  terms, 

—  2  sin8  b  sin8  c  cos3  A  —  3  sin  b  sin  c  cos  #  cos  c  cos8  y4  ) 

-f-  (i  —  cos2  b  cos8  c  -j-  sin8  #  sin8  ^)  cos  ^4  .  -j-  sin  b  sin  c  cos  b  cos  £  =  o.  f 

The  coefficient  of  cos  ^  reduces  to  sin'  b  -f-  sin8  r,  and  dividing  through  by  —  2  sin8  b  sin8  c 
we  have 

cos3  A-\-%  cot  #  cot  *:  cos8  A  —  %  (coseca  c  -\-  cosec8  b)  cos  A  —  %  cot  b  cot  c  =  o;  l 
cos3  *  -f  f  tan  d  tan  Z  cos8  /  -  \  (sec8  Z  +  sec8  d)  cos  /  -  £  tan  ^  tan  Z       =  O.  j 

88.  The  equation  in  terms  of  h,  L,  d\  a,  b,  c. 


T       /^..W          U    *.-..    i.-  /i          COS  «  —  COS   ^  COS 

In  (242),  substituting  cos  A  = 


sin  b  sin  c 
performing  the  operations,  collecting  terms,  and  simplifying,  we  have 

s  ,  sin8  c  -f-  sin8  b  •  \ 

cos  a  —  f  cos  b  cos  c  cos8  a  --    -J  --  cos  «  +  £  cos  £  cos  <:  —  o  ;   | 

V  (246) 

sin3  h  —  |  sin  d  sin  Z  sin8  h  —  \  (cos8  Z  +  cos8  d}  sin  ^  +  $•  sin  L  sm  d  —  o.    ) 


AZIMUTH.  53 

89.  The  equation  in  terms  of  Z,  L,  h  ;  B,  c,  a. 
Changing  the  sign  of  the  second  term  of  (233), 

cot  t  -f-  tan  h  sin  Z  cos  L  =  o  ;  ) 
cot  A  -j-  cot  a  sin  B  sin  c  =  o.  J 

Multiplying  by  sin  B,  sin  B  cot  A  -j-  cot  0  sin8  .Z?  sin  £  =  o.  ........  (248) 

Substituting  (235)  in  (248),  and  reducing, 

cosa  B  -f-  cot  c  tan  #  cos  B  —  2  =  o  ;  ........  (249) 

cos9Z-{-tan  jr  cot  h  cos  Z—  2  =  o  .........  (25°) 

90.  The  equation  to  the  projection  of  the  curve  of  equal  numerical  errors,  contrary  signs. 
Substituting  in  (250)  from  art.  55,  as  required,  and  reducing,  we  have 

-2  =  0  ...........     (250 


2x*  +  2x"  =  o  ;  ......    (252) 

or,  rearranging,        y  -f  3^y  +  2;r4  +  2Ay  +  2AA-jj/  —  y  —  2^r2  =  o  ......     (253) 

For  analysis  see  art.  IO2. 


PART  VII. 

ANALYSIS    OF    THE    EQUATIONS    TO    THE    LOCI,    AND    THE 

TRACING  OF  THE   CURVES. 


91.  To  Lieutenant  H.  O.  Rittenhouse,  U.S.N.,  the  writer  is  greatly  indebted  for  assistance, 
both  mathematical  and  constructive,  in  analyzing  the  equations  and  tracing  the  loci.  In  the 
progress  of  delineating  the  latter,  Lieutenant  Rittenhouse  gave  much  aid  in  removing  diffi- 
culties encountered.  The  drawings  for  the  accompanying  plates  were  made  by  him,  and  they 
represent  the  curves  faithfully  for  the  particular  latitudes  given,  to  serve  as  illustrations. 

For  the  determination  of  the  most  favorable  and  the  least  favorable  conditions  for  obser- 
vation of  the  star,  by  finding  the  algebraic  maximum  and  minimum  effects  of  the  errors  in  the 
data,  the  writer  claims  originality.  To  the  fact  that  text-writers  have  been  content  to  ascer- 
tain (?)  by  inspection  of  the  first  differential  of  the  particular  equation  employed  in  the  prob- 
lems for  time,  latitude,  and  azimuth  what  conditions  are  the  best, — is  due  the  fallacy  of  ac- 
crediting the  prime-vertical  the  place  of  best  observation,  when  error  in  h  and  error  in  t  are 
concerned,  in  the  altitude-azimuth  and  the  time-azimuth. 

Having  undertaken  this  work,  the  problem  of  finding,  in  any  given  case,  the  best  position 
for  observation  demanded  the  equation  in  /,  L,  d,  or,  as  an  alternative,  in  h,  L,  d,  for  L  and  d 
must  be  known  and  used.  By  means  of  these  two  equations  considerable  progress  was  made 
in  describing  the  curves :  where  one  equation  failed  to  clear  some  doubtful  point,  the  other 
would  sometimes  come  to  the  rescue.  But,  still,  there  always  remained  some  points  in  ob- 
scurity; inciting  guesses  where  reason  failed:  some  guesses  being  very  good,  as  ultimately 
proved ;  others  as  bad  as  could  be.  All  the  curves  had  been  defined  (?)  by  these  equations, 
and  the  writer  was  continuing  a  great  amount  of  "  dead  work"  among  them,  in  the  hope  that 
some  one  curve,  at  least,  simpler  than  its  fellows,  would  prove  accommodating — when  it  oc- 
curred to  him  that  the  tracing  of  the  curve  need  have  no  dependence  on  the  problem  of  find- 
ing, in  a  given  case,  the  time  to  observe  at,  or  the  altitude  to  observe ;  but  that,  in  a  given 
latitude,  by  varying  the  altitudes  for  points  of  the  curve,  the  corresponding  azimuths  might 
be  found  and  therefore  the  curve  could  be  traced.  Consequently,  the  equations  were  de- 
rived in  terms  of  Z,  L,  and  h  ;  and  it  was  then  discovered  that  these  equations  could  be  easily 
transformed  into  the  equations  in  terms  of  x,  y,  and  one  arbitrary  constant,  tan  Z,  for  refer- 
ring the  projection  of  the  curves  to  plane  rectangular  co-ordinates.*  If  this  were  not  accom- 
plished, still  the  equations  in  Z,  L,  h,  were  much  more  useful  than  the  others,  for  they  re- 

*  See  foot-note  to  art.  37. 


AZIMUTH.  55 

ferred  to  the  more  simple  system  of  spherical  polar  co-ordinates  ;  and,  at  the  same  time,  in 
general,  they  dropped  one  degree  lower  than  those  in  t,  L,  d  and  h,  L,  d. 

The  equations  to  the  projection,  once  obtained,  the  tracing  of  the  curves  became  com- 
paratively easy,  notwithstanding  some  of  the  equations  are  of  high  degree  and  involve  some 
nice  points.  It  is  not  believed  that  all  these  points  have  been  discovered,  and  the  writer  will 
be  gratified  to  see  newly  discovered  truths  respecting  the  curves  presented.  Considered  as 
plane  curves,  simply,  —  losing  sight  of  the  problem  from  which  they  are  evolved,  —  they  possess 
very  interesting  features.  Based  on  the  same  principles  giving  these  curves,  a  great  number 
of  others  may  be  discovered  by  considering  all  manner  of  variations  in  the  astronomical  tri- 
angle —  outside  of  the  utility  problems.  (For  this  subject,  see  appendix.) 

Each  curve  is  given  for  latitudes  30°,  45°,  60°,  thus  fairly  showing  the  alterations  of  form 
between  these  stages  as  the  curve  gradually  changes  between  the  limiting  latitudes  of  o°  and 
90°.  In  the  growth  of  the  curves  few  unlooked-for  changes  of  form  occur.  In  No.  I  and  No. 
4  some  surprises  are  met  with.  In  No.  4,  apart  from  analysis,  the  change  in  form  between 
30°  and  60°  of  latitude  graphically  hints  at  a  great  change  in  higher  latitudes. 

92.  Locus  No.  i.  Alt.-azimuth,  error  in  h.  Articles  43  and  51  to  56.  Branches  —  I. 
Algebraic  max.  and  min.,  giving  numerical  min.  2.  Absolute  min.,  curve  of  elongations.  3. 
Absolute  max.,  the  meridian. 

1st  Branch. 

(d]  By  (120),        /  +  4*y  +  s*y  + 


-f  4A/  +  ioA*y  +  8A*y  +  2  A*8 
+  4A2/  -  2/  -f 
—  4  A/  —  6A*y 

+/  +  2x*y  =  o  .................     (254) 

/ 
From  terms  of  highest  degree,  y  (j?  +  •*")"  (y  -|-  2x*)  =  o  ..........     (,255) 


Hence,  there  exists  an  infinite  branch  having  an  asymptote  parallel  to  X\  and  no  other  real 
infinite  branch. 

Coefficient  of  highest  power  of  x\s2y-\-  2A ; 


. ' .  y  =  —  A  for  the  asymptote, (256) 

which  is  the  projection  of  the  parallel  of  declination,  —  d  =  L (257) 


56  AZIMUTH. 

For  —  y=  tan  L,  and  disregarding  sign  of  y,  let  ?  =  arc  from  the  origin  to  the  point  a 
where  the  asymptote  cuts  the  meridian,  Y  ;  then 

tan  \z  —y  —  tan  L,  .'.  z  —  2L  ..........     (258) 

.  *  .  z  =  L  -\-  L.     On  Y  the  arc  from  the  origin  to  the  equator  =  L  ;   from  equator  to  the 
point  a,  the  arc  =  dec.  of  point  a  =  the  remaining  L,  .  *  .  resuming  sign,—  d  =  L. 
(b]  For  points  of  locus  on  Y,  put  x  =  o,  .  •  . 


o;      ......     (259) 

=  o  ..........   (260) 

First  factor  gives  y*  =  o,  the  origin. 


Second  factor  gives  y  =  —  A  ±  4/A2  -f-  i (261) 

N.  and  S.  poles,  P  and  P'.     (Imaginary,  see  (/)). 
(c)  For  points  on  X,  put  y  =  o. 


=  o; 

.  • .  x*  =  o,  the  origin  ;  x*  =  ±  i,  E.  and  W.  points (262) 

(d)  For  form  at  origin,  from  terms  of  lowest  degree : 

y  -f-  ^y  =  o,         or        y  (y  -f-  2x*}  =  o,  (imaginary) (263) 

But  from  2yx*  —  2Ax*  =  o (264) 

y  =  tv? (265) 

V' 


And  the  curve  is  of  the  form 


'g-  4). 


(e)     Form  of  curve  at  E.  point. 

Moving  origin  to  (i,  o)  the  coefficient  of  x  is  found  to  be  4A;  the  coefficient  of  y, 
4  A' 


.  •  .  y  =  --    —  T-a  x  is  the  tangent  at  E  ........    (266) 

I  -f-  A 

_A 

Similarly,  y  =       .    /rs  is  the  tangent  at  W  ..........     (267) 


AZIMUTH.  57 

(/')  To  ascertain  the  character  of  the  locus  at  P  and  P'  ;  moving  origin  to  P,  putting  for 
y,y  —  A  -(-  4/A3  -f-  I,  the  coefficient  of  y  vanishes.  We  have  for  determining  the  sign  of  the 
coefficient  ofy, 

|2A4  +  3A1  -f  i  -  2A3  |/A2+  i  -  2A  4/Aa  +  i  ;  } 


similarly  for  x\  |2A4  +  3A2  +  i  -  2A3  l/A^+1  -  2A 

The  signs  of  these  two  terms,  therefore,  will  be  always  the  same,  and  the  form  at  P  is  imag- 
inary. P  is  therefore  a  peculiar  point,  and,  from  the  symmetry  of  the  locus,  P'  must  also  be 
a  peculiar  point. 

(£•)  Though  no  real  branch  of  this  curve  passes  through  P  or  P',  curves  to  follow  do 
pass  through  these  points;  therefore,  to  prove  that  the  N.  and  S.  poles  -are  given  by 
y  —  —  A  ±  4/A2  -\-  i,  we  have  for  P,  the  N.  pole,  co-L  =  arc  from  the  origin  (zenith)  to  the 
pole.  By  stereographic  projection,  if  r  is  the  linear  distance  from  origin  to  P, 

then  r  =  tan  \  co.-L  ...........     (268) 

i  —  cos  co-Z,         i  —  sin  L  ,   .,  N 

By  trig.,  tantco-£=  =  --  ...........    (269) 


sin  L          /sin2  L  -f-  cos*  L  .      . 

1     '         5-r-       •    •    •    (27°) 


-  Y—         . 
cos  L 

Similarly  for  P', 


I  —  sin  L  .      . 

y  ...........    (271) 


_       i  _|_  sin  L 

or        -j  =  A  +  fA2-f  i=  ~;     -    -    (272) 


r'  =  distance  from  origin  to  P'  =  tan  •£  (90°  +  L). 

i  —  cos  (go0  -r  L       i  +  sin  L 
By  trig,  tanl(90°  +  £)=  "-'      .....     (273) 


r'=-j  .............     (274) 

(^)  The  curve  does  not  cross  its  asymptote,  for,  combining  (256)  with  (254),  we  have 

(A'  +  2AX  +  2A(A'+i)y-fA3(A'  +  i)':=Q,      .....     (275) 

which  gives  imaginary  roots  only. 


58  AZIMUTH. 

(/)  For  limiting  form  of  curve,  L  =  o,  put  A  =  o  in  (254), 

y  =  o  divides  out  ;    ........... 

.-.  axis  of  X,  the  prime-vertical,  is  a  part  of  the  locus.     There  remains 

y  +  4*y  +  5  *y  +  2*6  —  2/  —  2^y  +  y  +  2**  =  o  .....  (277) 

Infinite  branches  are  imaginary  and  form  at  origin  imaginary. 
If  y  =  0,  S  =  O,     ................     (278) 

and  x  =  V—  i,  imaginary.  ...........     (279) 

If  x  =  o,  y  =  o,     ................     (280) 

and  (y  —  i)a  =  O,        #  =  ±  I,  double  point  at  N.  and  S.      .     .     (281) 

Moving  origin  to  N.,  putting  y  =  y  -f-  r>  the  f°rm  *s  given  by 

*>+y  =  o;    ............     (282) 

.'.  imaginary  branch  at  N.  point,  the  same  at  S.  point  ;  N.  and  S.  are  isolated  points. 

Arranging  (277)  as  a  cubic  in  x*,  we  have 

+  (4y  -2y  +  2)^+y(y-2/+i)  =  o  .....  (283) 


These  coefficients  are  all  positive,  therefore  x*  can  have  no  positive  root,  and  therefore 
x  can  have  no  real  root.  The  locus  is  therefore  imaginary,  excepting  the  line  y  —  o  (that  is 
X,  the  prime-vertical  which  is  the  equator),  and  the  peculiar  points  at  zenith  and  N.  and  S. 

(k)  For  limiting  form  L  =  90°.     Put  A  =  oo  in  (254)  and  we  have 


=  o; (284) 

/.  y  =  o,  giving  X,  the  prime-vertical ; (285) 

(x* '-}-  y)s  =  o,  giving  origin,  zenith,  a  double  conjugate  point.     .     .     .     (286) 

(/)  Not  considering  detached  points,  it  is  seen  from  (/)  and  (K)  that  starting  with  Lat.  =  o, 
the  axis  of  X  is  the  locus  ;  and  ending  with  Lat.  =  90°,  again  the  axis  of  X  is  the  locus; 
while  between  these  limiting  latitudes  there  is  always  one  branch  passing  through  the  origin 


AZIMUTH. 


59 


and  E.  and  W.,  and  always  having  an  asymptote/  =  —  A.     On  the  sphere  it  is  a  continuous 
branch  through  zenith,  E.,  nadir,  W.,  to  zenith. 

The  question  arises  :  How  does  the  curve  change  its  shape  with  the  change  of  latitude, 
so  that  it  shall  return  to  its  original  form? 

[NOTE. — For  the  following  elucidation  and  the  determination  of  the  envelope,  the  writer  is  much  indebted  to 
Lieutenant  Rittenhouse  and  Professor  Hendrickson.] 

The  equation  (254)  to   the  curve  contains  A  and  A2.     Arranging  it  as  a  quadratic  in  A, 
we  have 


=  o. 


(287) 


Now,  for  any  values  of  x  and  y,  two  values  of  A  can  be  found  to  satisfy  the  equation. 
That  is,  suppose  the  curve  drawn  for  a  given  latitude,  take  some  point  on  it,  and  this  reason- 
ing declares  that  for  some  other  latitude,  also,  the  curve  will  pass  through  the  chosen  point. 
The  fact  that  the  curve  returns  to  its  initial  shape  while  the  latitude  increases  to  90°  is  prim  a- 
facie  evidence  that  the  curve  in  some  way  reaches  a  limit  and  then  returns.  This  taken  in 
connection  with  the  quadratic  character  of  A  points  to  the  envelope  of  the  curve  as  a  means  of 
obtaining  the  limit  sought.  The  envelope  is  obtained  from  the  condition  fi*  =  ^AC  (in  the 
general  quadratic  equation)  ;  that  is,  it  is  the  condition  that  will  make  A  have  equal  roots  in 
the  locus  (287). 

Squaring  the  coefficient  of  A  and  putting  the  result  equal  to  four  times  the  coefficient  of 
A3  into  the  absolute  term  and  simplifying,  we  obtain 


4/V  -f  is/*4  -f- 
Factoring, 


—  x"  -f  8j>V  +  26/V  + 


-f 


) 
-  x?  =  o.  i 

+  ^]=0..      (289) 


(ni)  The  equations  given  by  the  several  factors  of  (289)  make  up  the  envelope.  The  first 
factor,  x*,  gives  the  line  Y  ;  the  second  factor,  (x*  +y)2,  gives  the  point  (o,  o),  the  origin  : 
neither  is  of  use  to  us.  But  the  remaining  factors  give  a  locus  shown  in  Fig.  5. 

[NOTE.  —  The  part  for  the  3d  factor  alone  will  be  discussed.] 

From  which  we  see  that  the  line  ZME  may  form  a  boundary  within  which  the  curve  No.  I 


FIG.  5. 


may  pass  through  its  phases.     All  facts  ascertained  respecting  the  curve  develop  nothing  to 
conflict  with  such  a  conclusion,  and  all  facts  are  satisfied  by  such  conclusion. 


60  AZIMUTH, 

(ri)  Taking  the  3d  factor  of  (289), 

(*>+/)  (27 +  *)  + 27 -*  =  a      .     .     .     .     .    *    i    .     (290) 
The  only  infinite  branch  is  given  by         2^-j~;c  — °» (29!) 

x 

and  its  asymptote  is  jp  = , (292) 

2 

since  there  are  no  terms  of  the  degree  next  to  the  highest. 
For  points  on  X  and  Y, 

Put  y  —  o,        x3  —  x  =  o,       x  (X*  —  i)  =  o,        .#  =  o,  origin  ;  \ 

x-  ±  i,  E.  and  W.     j    '     (293) 


Put  x  =  o,      2y*  4-  2y  =  o,        r  ( Vs  4-  i)  =  o,        y  =  o,  origin  ; 

>     (294) 
^  =  l/_  i,  imaginary. 


For  tangent  at  origin,  27  —  JT  =  o,        y  —  ~  ........     (295) 

2 

For  tangent  at  E.  point,  put  x-\-  I  for  ;r. 

x 

Lowest-degree  terms  are  4/-f~2;tr  —  °I     •'•J/=  --  .......     (29^) 

2 

The  curve  crosses  its  asymptote  only  at  the  origin. 

A 

(o)  By  (266)  the  tangent  to  the  curve  No.  I  at  E  is  found  to  bej  =  —          —»-;•*'.     This 

I  -f-A 

expression  has  a  limiting  value  i  when  A  =  i,  L  =  45°.     Now  this  limit  of  the  inclination  of 
the  curve  at  E  is  exactly  equal  to  the  inclination  of  the  envelope  at  E  by  (296). 

(/)  To  find  the  maximum  ordinate  of  the  envelope  (for  the  point  M). 

The  equation  to  the  envelope  (290)  expanded  is 


-f  */  -f  2;r>  +  *8  -f  2j/  —  *  =  0  .........     (297) 


dx  6/  +  zxy  -\- 

For  maximum  ordinate  we  have  -~  =  o.     Putting  the  numerator  =  o,  we  have 


f  +  W  +  3**  -  i  =  o  ...........     (298) 


AZIMUTH.  61 

Solving  (297)  and  (298)  for  x  and  y,  we  obtain  from  (298) 


y=  —  2x±  Vi  -\-x* (299) 

Substituting  (299)  in  (297),  we  have,  after  squaring  and  simplifying,  the  cubic 

2$xe -\-  \6x*  +  I2x*  —  4  =  0; (300) 

From  which  an  approximate  root  is  found 

^  =  .238126  +  ;        .-.*  =  0.483  5422+ (301) 

Substituting  (301)  in  (299)  we  find 

y  =  0.1468784  +  for  max.  ordinate (302) 

(q)  To  ascertain  the  character  of  the  curvature  of  No.  I  at  E. 

In  (254)  move  the  origin  to  (i,  o). 

Ax 
The  terms  of  first  degree  give  only  the  line  y  = : — v-j,  (266).     Taking  the  terms  of 

1  H-  A 

second  degree  in  connection  with  those  of  the  first  degree,  we  have  the  conic  whose  curva- 
ture near  the  origin  is  similar  to  that  of  the  curve.     The  equation  of  the  conic  is 

A/  -f  8  (i  +  Aa)  xy  +  9 A*2  +  2(1+  A>  -f  2 A*  =  o.  (hyperbola).    .     .     (303) 
If  y  =  o,  we  have  gAx*  -f-  2 Kz  =  o ; 

whence  for  all  values  of  A,        x  =  o,        x  =  —  f (3°4) 

If  x  =  o,  we  have  Ay*  -f-  2  (i  +  A") y  =  o; 

2  (i  +  A9) 
y  =  0,        y= J i (305) 

The  second  value  of  y  always  negative  for  positive  values  of  A  and  never  less  numerically 
than  4. 

For  asymptotes, 


{-4(A'+  i)  -  l^i6A4  +  23A'+  i6|  x  -  (4A*  +  8Aa+  5) 

y  = ~K~ 


|-  4(Aa  +  i)  +  VI6A*  +  23A'+  i6|  x  f  (4A* +  6Aa  +  3)  .     n. 

y  = ^ .  .     .     .     (307) 


62  AZIMUTH. 

(306)  and  (307)  are  the  asymptotes  of  the  form 

c  —  —  mc      £  m       m          and 


These  asymptotes  always  intersect  in  the  2d  angle  ;   and  since  the  curve  always  goes 
through  the  origin  and  has  the  two  negative  intercepts,  the  curvature  at 
E  is  always  of  the  form  shown  in  Fig.  6.  —  ^^ 

(r)  The  conclusions  regarding  the  envelope  are  supported  by  accurately  •  - 
constructed  curves  for  lat.  30°,  45°,  and  60°,  and  also  for  a  high  latitude, 
A  =  4',  lat.  about  75°.  For  L  <  45°  the  curve  is  not  tangent  to  the 
envelope  ;  L  =  45°  it  is  tangent  at  E.  point  ;  L  >  45°  it  is  tangent  at  some 
point  S.  of  the  envelope,  but  the  curvature  diminishes  at  E.  As  L  increases,  the  point  of 
tangency  S.  moves  towards  Z,  and  the  curve  flattens  more  and  more  rapidly  until  it  coincides 
with  the  prime-vertical.  From  a  mathematical  point  of  view  the  latitude  to  which  most  sig- 
nificance should  be  attached  would  appear  to  be  that  at  which  the  curve  ceases  to  swell 
at  E.  and  begins  to  swell  towards  the  origin,  that  is,  lat.  45°.  But,  for  our  purposes,  the  lati- 
tude that  gives  the  broadest  swell  to  the  curve  is  significant,  for  the  curve  will  then  have  the 
widest  departure  from  the  prime-vertical. 

The  curve  constructed  with  any  latitude  will  have  its  own  greatest  ordinate.  In  the 
system  of  curves  the  maximum  of  these  greatest  ordinates  must  be  the  maximum  ordinate  of 
the  envelope  ;  and  the  curve  will  be  tangent  to  the  envelope  at  the  extremity  of  this  ordinate. 
The  writer  has  not  yet  been  able  to  find,  analytically,  the  exact  latitude  which  will  give  this 
maximum  ordinate. 

The  approximate  numerical  value  of  the  latter,  and  that  of  the  corresponding  abscissa 
(found  in  (/)),  when  substituted  in  the  equation  of  the  curve  expressed  as  a  quadratic  in  A, 
(287),  will  give  an  approximate  value  of  A,  whence  the  desired  latitude. 

Assuming  the  roots  equal  (the  numerical  solution  verifies  the  equal  roots),  we  have 


A     __    V   -^ 

Substituting  the  values  of  x  and  y,  the  computation  gives 

A  =  1.40242  =  (tan 


L  =  54°  30'  33-"  ' 

93.  Locus  No.  i.     2d  Branch. 

(a)  By  (123),  /  +  *>  +  2A/-f2A**-.y  =  o (310) 

By  terms  of  highest  degree  y(y*-\-x*}  = 

one  infinite  branch  whose  asymptote  is  parallel  to  X. 


AZIMUTH 
By  highest  power  of  x,        y  -J-  2A  =  o ;        /.  y  —  —  2A  .     . 


gives  asymptote.  The  curve  does  not  cross  its  asymptote.  This  asymptote  cuts  the  meridian, 
Y,  at  twice  the  distance  from  the  origin  that  the  asymptote  to  the  1st  branch  of  this  locus 
cuts  it,  for  by  (256)  y  =  —  A. 

(b)  To  find  where  the  curve  cuts  Y,  put    x  =  o ; 

/.  y  -f-  2A/  —  y  =  o,        or        y  (y9  -|-  2 Ay  —  i)  =  o ; 

y  =  o  origin,      and      y  =  —  A  ±  4/F+~Aa ;  P  and  P' (313) 

To  find  points  on  X,  put  y  =  o;        .-.xy  =  o (314) 

Tangent  at  origin,    y  =  o,     axis  of  X.  > 

Tangents  at  P  and  P',  parallel  to  X.       )'  •     (3*  5) 

(c)  Limiting  form  for  L  =  o,  A  =  o. 

-  y  =  o,        or       y  (y*  -J-  &  — *  J)  =  °  '•> 


y  =  o,  axis  of  X,  prime-vertical  ; 


xs  o      ,  prme-vertca  ;          ) 

r  ••••('*  i  ( 

=  i,  primitive  circle,  horizon.  )  ' 


y  -f-  X* 
(d?)  Limiting  form  for  L  =  go,  A  =  oo  . 

2Ay  +  2A^a  =  o,        y  -f-  #*  =  <>•  origin,  zenith  ......     (317) 

The  3d  branch  the  meridian,  axis  of  Y. 

94.  Locus  No.  2.  Alt.  -azimuth,  error  in  L.  Articles  44  and  60.  Branches  —  I.  Absolute 
min.,  the  six-hour  circle.  2.  Absolute  max.,  the  meridian. 

1st  Branch.  —  Though  a  great  circle,  easily  constructed,  yet,  as  interesting,  a  brief  analysis 
is  given. 


By(l26),  /  +  2Aj-f^—  l  =  o  ...........  (318) 

There  is  no  infinite  branch. 

Points  on  X,  x=  ±  I,  E.  and  W.  ;     .     ...........  (319) 

Points  on  Y,  y  =  —  A  ±  V\  -+-  A2,  P  and  P'  .........  (320) 


64  AZIMUTH. 

Tangents  at  E.  and  W.,  j  =  -r-,        y  =  —  -^ .    .    0    .     .    (321) 

Tangents  at  P  and  P'  parallel  to  X. 

94a.  Locus  No.  3.  Altitude-azimuth,  error  in  d.  Articles  44  and  61.  Branches  the 
same  as  in  No.  2,  but  the  six-hour  circle  a  curve  of  true  max.  and  min.,  giving  numerical  min. 

95.  Locus  No.  4.  Time-azimuth,  error  in  t.  Arts.  45  and  62  to  67.  Branches — i.  Alge- 
braic max.  and  min.,  giving  numerical  min.  2.  Absolute  min.,  curve  of  q  =  90.  3.  Curve  of 
algebraic  max.  and  min.,  giving  generally  numerical  max.,  the  meridian. 


(a)  By  (155),     /  +  2*y-f*>  +  A/  +  2A*'/  +  A*4  -  A/-  Ax*+y  =  o.    .     (322) 
Highest-degree  terms,  y  (y  -}-  ^r2)2  =  o,      ...........     (323) 

gives  one  infinite  branch  only,  that  having  an  asymptote  parallel  to  X.     The  coefficient  of 
highest  power  of  x, 

A  -\-y  =  o  ;     /.  y  =  —  A,  asymptote,  ........     (324) 

the  projection  of  the  parallel  of  declination,  —  d  =  L  (see  art.  92  (#)). 

(b)  Points  on  X. 


If  y  =  o,  x*  (l  —  *a)  =  0,  x*  —  o,  origin  ; 

x  =  ±  i,  E.  and  W 


Points  on  Y. 


\ 
.  \ 


+  A/  — 
=  o,  origin  ;         and        y  -f-  Ay  —  Ay  -j-  i  =  o, 


_?.}       •    '     (326) 
—  r  ,  » 


(See  (^  to  (0-) 

(<:)  Tangent  at  origin,  axis  of  X. 

Form  of  curve  at  origin,         y  =  A*",  same  as  in  No.  i  (see  art.  92  (d}}  ......     (327) 

(d)  For  tangent  at  E. 


,   g, 


/.  for  tangent  at  E.,  y  =  —  A*  ;  } 

similarly,  tangent  at  W.,  y  =       Kx.   \ 


AZIMUTH.  65 

(e)  For  points  on  Y  other  than  the  origin.     Returning  to  (326),  we  have 

y  +  Ay  -  A>/  +  1  =  o  ...........  (330) 

Not  having  been  able  directly  to  factor  (330),  recourse  is  had  to  the  equation  to  the  curve  on 

A  sin  h  cos  h 

the  surface  of  the  sphere,  (153),  cos  Z  =  -       .  a  ,    .     The  condition   required  to  give 
,  i.  ~]    sin   /? 

points,  other  than  the  origin,  on  Y  is  Z  —  o  or  180°  ;  /.  cos  Z  —  ±  i,  and  we  have 

±(i  -j-sin1^)  =  A  sin  h  cos  h  ..........     (331) 

(/")  From  (330)  we  see  that,  with  L  =  o,  y  is  imaginary,  and  no  point  on  Y  ;  and  with 
L  =  go0,  when  A  =  oo,  we  have_^  (y  —  i)  =  o;  y  =  o  and  y  =  ±  I  ;  .'.  the  two  points  on 
the  primitive  at  its  intersections  with  Y  ;  that  is,  on  the  sphere,  at  the  intersections  of  the 
meridian  with  the  horizon,  giving  h  =  o  for  each  point. 

Now,  by  (331),  for  any  permissible  values  of  L  <  90°  to  give  points  on  the  meridian,  sin 
h,  hence  h  also,  must  be  positive  for  cos  Z  =  -f-  i  ;  and  negative  for  cos  Z  =  --  i  ;  that  is, 
-\-  y  cannot  exceed  unity,  and  —  y  cannot  be  less  numerically  than  unity. 

(<§")  Squaring  both  members  of  (331),  we  have 

I  +  2  sin2  h  -f  sin4  h  =  A2  sin2  h  cos2  h  =  A2  sin2  h  (i  —  sin2  /i)  =  A8  sin8  h  —  A2  sin4  h,  (332) 

2  —  A2  i 

whence  sin4  h  -f  -pqp^i  sin8  h  =  -    t    ,    A»  ........     (333) 

Solving  (333)  as  a  quadratic, 

(A8  -  2)  ±  A 

- 


(334) 


.     ,  A8  -  2  ±  I/A8  -  8  ,      . 

•••  sm  h  --        ±  -  —     -       .......  (335) 


By  trig.,  cos  #  =  —  -j  —  -  —  a~r",  which  by  the  principles  of  stereographic  projection  gives 

I    —  r"    tclll       0*3* 


i  —  y 

coss  =--     i;    ...........     (336) 


whence,  taking  the  positive  sign  of  the  radical  in  (335), 

A2  -  2  -f  A  ^A8  -8 


i  -  /        j 

r+?  = 


66  AZIMUTH. 

squaring  both  members  of  (337),  clearing,  and  dividing  by  the  coefficient  of  y\ 


6Aa      2A  4/A3- 


4  +  A3- 


—  I. 


(338) 


Solving  (338)  as  a  quadratic, 


_  3A8+  A  I/A'  -  8  ±  |2  (A4  +  A3  VAa  -  8  -  Aa  +  A  i/ATTg  -  2)|* 

4  +  A'  -  A  |/A^8 


5     '     (339' 


whence  the  two  values  of  j,  rejecting  the  inadmissible  signs,  to  conform  to  the  conditions 
in  (/),  are 


y  = 


3A2  +  A  i/A3  -  8  -  2[2  (A4  -f  A3  V Aa  -  8  -  A3  -f  A  V A3  -  8  -  2)]*  )  * 


A2-A4/A3-8 


}*(340) 


3  Aa  +  A  VA*  -  8  -f  2  [2  (A4  +  A*  i/A3"^  -  A3  +  A  l/A3"^  -  2^1* 
-  ~ 


(/)  There  remain  two  other  values  of  y  found  by  taking  the  negative  sign  of  the  radical 
in  (335)>  whence 


TT?^    ( 


A3  -  2  -  A  V  A3  -  8 
2(1+ A3) 


(342) 


Solving,  as  in  (K),  rejecting  inadmissible  signs,  we  have 


3A'  -  A  I/  A3  -  8  -  2  [2(A4-  A3  VA*  -  8  -  A'-  A 


8  -  2)] 


*     * 


(  4  _j_  A3  +  A  V Aa  - 

(  3Aa  -  A  t/A'— 8  +  2  [2 (A4 -  A3 


(343) 


8  -  A*  -  A  4/A3  -  8  -  2)] 


*  )  * 

-  [     (344) 


(^)  It  is  obvious  that  these  values  of  y  are  imaginary  for  any  value  of  A*  <  8  or  A  <  2  t/2 
[also  seen  from  (335)].  Hence,  with  the  lower  latitudes,  the  curve  does  not  touch  the  me- 
ridian at  any  point  other  than  the  origin.  For  the  latitude  whose  tan  is  2  4/2~  we  have  two 
pairs  of  equal  roots,  for  then  (340)  and  (343)  become  identical,  and  (341)  and  (344)  identical. 
What  have  been  in  the  lower  latitudes  continuous  branches  of  the  curve  on  each  side  of  the 
meridian,  passing  through  the  zenith  and  E.  and  W.  points  (Fig.  6),  at  the  instant  A  =  2  4/2, 
cross  on  the  meridian.  (Fig.  7,  for  one  pair  of  equal  roots  above  the  horizon.) 


AZIMUTH. 


67 


FIG.  6. 


FIG.  7. 


FIG.  8. 


Similarly  below  the  horizon  for  the  other  pair  of  equal  roots. 

But  when  A  becomes  greater  than   2  4/2  by  the   smallest    increment,  the  equal  roots 
separate,  and  we  have  two  different  roots,  and  the  branches  of  the  curve  separate  as  in  Fig.  8. 
(/)  To  investigate  for  the  points  on  the  meridian,  Y,  where  the  equal  roots  occur: 
I st.   The  altitude  of  the  point. 


In  (334)  substitute 


A  =  2  4/2,         A2  =  8; 


whence 


sin'  h  = 


sin  h  =  ±  V$  =  ±      =. 


(345) 


Hence  two  branches  of  the  curve  cross  each  other,  above  the  horizon,  on  the  meridian, 

at  an  altitude  whose  sine  equals  -f-  — =;  the  zenith-distance  of  this  point  having  the  direction 

v$ 

towards  the  elevated  pole.     And  the  lower  branches  cross  on  the  meridian  at  a  negative  alti- 
tude numerically  equal  to  the  positive,  reckoned  towards  the  depressed  pole. 

(;»)  2d.    The  declination  of  the  point. 

To  find  the  parallels  of  declination  passing  through  these  points. 

Taking  the  equation  in  terms  of  /,  L,  d  (139),  the  second  factor  gives 


2A  cos  d  i 

cos  t : — — -  cos  t  -\-  _  ^  j  —  A2  =  o ; 


sin  d 


cos2  d 


(346) 


cos  /  =  ±  i,      whence       i 


2A  cos  d          i 

sin  d          cos''  d 


-A2=o;.     .     .     .     (347) 


2A  cos  d  i 

T~       A  I 

-\-  -  ,      —  .TV    - 


sin  d 


cos2  d' 


(348) 


Substituting 


A  =  21/2, 


4  4/2  cos  d 


cos8  d' 


(349) 


68  AZIMUTH. 


32  cos*  d  14  i 

Squaring  both  members,  -^^r-  =49-^7^  H-^r 


clearing  of  fractions,  32  cos6  */=  49  sin2  ^  cos4  d—  14  sin"  <a?  cos8  d-\-  sin"  ^;    ....  (351) 

substitute  I  —  cos3  d  for  sina  d, 

32  cos"  d  =  49  cos4  */  —  49  cos8  */  —  14  cosa  d-\-  14  cos4  rf-|-  i  —  cos2  d\  .     .  (352) 

.-.  81  cos'  d—  63  cos4  ^+15  cos2  */  —  i  =o;      ......  (353) 

cos'  d  —  %  cos4  </+  ^T  cos4  d  —  -fa  =  o,      .......  (354) 


giving  a  cubic  in  cos*  </. 

Roots  for  cos"  d  are  £,  £,  and  \. 

Since  dfis  limited  to  ±  90°,  the  negative  roots  for  cos  afare  inadmissible.  Both  of  the 
equal  roots  -\-  V  %  correspond  to  a  plus  declination,  and  both  to  a  minus  declination,  numeri- 
cally equal.  Now,  this  value  of  cos  d  or  sin/  is  exactly  equal  to  the  numerical  values  of  sin 
//  and  —  sin  h  by  (345). 

/.  sin/  =  sin  h  ;  and,  calling  /'  the  polar  distance  for  the  negative  declination,  h'  the 
negative  altitude,  sin/'  =  sin  //''  (numerically). 

Hence  for  A  =  2  V  2  two  branches  of  the  curve  cross  each  other  on  the  meridian  at  a 
point  midway  between  the  elevated  pole  and  the  horizon  ;  and  two  branches  cross  at  the 
middle  point  between  the  horizon  and  the  depressed  pole. 

Since  h  =  /  it  follows  that  z  =  d,  that  is,  the  arc  corresponding  to  -f-  y  has  the  same 
value  as  the  declination  of  the  body  that  crosses  the  meridian  at  the  point  where  equal  roots 
occur. 

Similarly  the  arc  corresponding  to  —  y  is  numerically  the  supplement  of  the  negative 
declination  of  the  body  crossing  the  meridian  below  the  horizon,  where  the  equal  roots  occur; 
that  is,  2'  =  1  80°  —  d'  (numerically). 

There  remains  the  root  -£  =  cos  d.  This  root  corresponds  to  both  -)-  d  and  —  d,  equal 
numerically  and  equal  to  the  latitude  giving  the  equal  roots.  That  is,  a  branch  of  the  curve 


passes  through  the  zenith  and  through  the  nadir.     For,  since  cos  d  =  £,  sin  d  =  Vi  —  £; 

/.  tan</=  34/f  =  t/8  =  2  t/2  ; (355) 

/.  tan  d  —  tan  L  =  A  =  2  i/2~. 

(»)  Although  the  expression  (326)  f  -f-  Ay  —  AJJ/  -f-  i  —  o  may  not  be  factored,  ap- 
proximate roots  may  be  found  for  different  values  of  A,  and  the  latitude  for  which  equal 
roots  occur  may  be  found  as  follows,  suggested  by  Lieutenant  Rittenhouse : — 


AZIMUTH.  69 

The  condition  for  equal  roots  in  a  biquadratic  equation, 


-f  *  =  o,      ........     (356) 

is  \ae  —  $d  +  y*\*  =  zftad*  -\-eff  +  t  —  ace—  2bcd\,  .....     (357) 


and  this  condition  applied   to  (326)  gives  A  =  2  1/2,  as  the  value  of  A,  for  which  equal  roots 
occur.     Found  by  substituting  in  (357), 

a=  i,        4$  =  A,        6c  =  o,        4d=  —  A,        e  =.  i. 
Substituting  this  value  of  A  in  (326),  it  will  factor,  and  we  obtain 


Hence  we  have  two  pairs  of  equal  roots,  and  the  four  roots  of  the  equation  are  all  im- 
aginary for  values  of  A  less  than  2  V2,  and  real  for  values  of  A  greater  than  2  1/2,  but  their 
numerical  values  are  unequal. 

These  facts  show  that  the  curve  touches  the  meridian  when  tan  L  —  2  V2,  and  that  it 

1/T—  i 
has  real  branches  ;  for,  moving  the  origin  to  y  =  -  -=.  —  ,  we  find,  for  tangents, 

V2 

y=±  ^x  .............    (359) 

These  tangents  also  show  how  the  curve  abruptly  changes  its  character  in   passing  a  critical 
value  of  its  parameter  A.     See  Figs.  6,  7,  and  8  in  (>£). 

(o)  As  a  matter  of  interest  the  approximate  values  of  the  latitude,  etc.,  for  the  occurrence 
of  equal  roots  may  be  found. 

Lat.  70°  32',  co-L  =  19°  28',         A  -p  =  35°  16,  z  =  d=  54°  44', 

-  d'  =  54°  44',        *'  =  (180°  -  d'}  =  125°  16',         -  h'  =  (180°  -/)  =  35°  16'. 

Conclusions,  looking  at  the  change  in  the  character  of  the  curve  in  high  latitudes. 

In  Lat.  70°  32',  the  most  favorable  position  of  the  star  whose  dec.  is  54°  44'  is  on  the 
meridian  at  lower  culmination,  as  far  as  possible  from  the  prime-vertical  in  bearing.  Where 
this  star's  parallel  of  declination  crosses  the  meridian  is  the  point  of  intersection  of  two  curves 
of  algebraic  max.  and  min.  ;  namely,  the  curve  under  present  discussion  and  the  meridian. 
It  has  been  asserted  that  the  meridian  is  generally  the  locus  of  numerical  max.  ;  though  the 
numerical  maxima  would  not  be  equal  numerically  for  the  two  culminations  of  the  body,  yet 
at  both  culminations  we  shall  have  maxima  as  compared  to  the  numerical  minima  on  the 
first  branch  of  curve  No.  4,  at  the  points  of  the  star's  crossing  it,  for  stars  whose  ±  d  <  L. 


7°  AZIMUTH. 

This  holds  good  for  all  such  stars  with  latitudes  less  than  70°  32'  (approx.).  With  higher 
latitudes  there  may  be  a  belt  (a  belt  above  the  horizon,  also  one  below)  crossing  the  meridian, 
within  which  stars  having  certain  declinations  between  the  limiting  declinations  of  the  points 
at  which  the  two  branches  of  the  curve  intersect  the  meridian,  will  not  touch  the  curve  of 
numerical  min.  For  such  stars  there  will  be  a  numerical  max.  and  a  numerical  min.  on  the 
meridian,  corresponding  to  algebraic  max.  and  min.;  for  there  exists  no  other  minimum  with 
which  to  compare.  From  (58)  we  see  that  the  numerical  min.  will  occur  at  that  culmination 
of  the  star  that  has  the  less  altitude,  whether  positive  or  negative.  But  the  sign  of  (58)  being 
negative,  the  numerical  min.  will  be  the  algebraic  max.,  regarding  the  error  in  t  as  positive. 
We  therefore  see  how  erroneous  may  be  the  statements  italicized  in  arts  5  and  9,  when  ap- 
plied to  observations  in  very  high  latitudes  ;  for,  with  declinations  permitting  the  star  to  cross 
the  first  branch  of  the  curve,  the  best  position  may  be  very  near  the  meridian  in  azimuth. 
(/)  For  limiting  forms  of  the  curve, 

L  —  90°,        A  =  oo. 
From  (322)  we  have  (x*  +  y)  (i  —  x*  —  y1}  =  o;     .........     (360) 


X*  -j~y  =  O,  the  zenith,  a  point  ; 
:=  I,  the  horizon. 


L  =  o,        A  =  o,        ^r>  +  2^y+/  =  o;       j(*'+y)°  =  o;  .     .    .     (362) 

y  =  o,  the  prime-vertical  ;     ) 
x*  +  y  =  o,  the  zenith,  a  point,    f  ' 

96.  Locus  No.  4.     Second  branch,  q  •=.  90,  for  absolute  min. 
The  same  as  in  Locus  No.  I,  art.  93. 

<)6a.  Third  branch.  —  The  meridian  for  algebraic  max.  and  min.  ;  numerical  max.,  for  all 
latitudes  below  that  for  which  tan  L  =  2  4/2,  at  both  culminations  of  the  star  ;  the  max.  at 
the  culmination  having  the  less  altitude,  whether  positive  or  negative,  being  the  less.  For 
higher  latitudes  than  tan  L  =  2  4/2,  stars  within  certain  declinations  will  have  one  numerical 
min.  and  one  numerical  max.  on  the  meridian,  and  will  not  touch  the  first  branch  (see  art.  95  (0)). 

97.  Locus  No.  5.      Time-asimuth,  error  in  L.     Arts.  45,  68  to  71.     Branches  —  I.  The 
horizon  for  absolute  min.     2.  The  meridian  for  absolute  min.     3.  The  branch  given  by  (179), 
for  algebraic  max.  and  min.,  giving  numerical  max. 

(a)        y  +  *y  -  *y  -  *°  +  2  A/  +  4A*y  +  2  A*> 

-  **  =  o. 


Highest-degree  terms  give          (y  —  x)  (y  +  x)(f  +  #*)'  =  O  ..........     (365) 

Hence  infinite  branches  having  asymptotes. 


AZIMUTH.  71 

To  find  asymptote  parallel  to  (y  —  x}  =  o  we  have  from  (364),  retaining  only  highest- 
degree  terms  and  those  of  the  next  degree, 


(366) 


Also, 

.*.  y  —  x  =  —  A        and        y  -f-  x  =  —  A  for  asymptotes.       .     .     .     (368) 

» 

There  are  no  parabolic  infinite  branches. 

(b)  Points  on  X.     Put  y  =  o. 

-x'-2x<-x*  =  o;        .-.*•(*'+ ,)'  =  o; (369) 

whence  x3  —  o,  origin,  zenith,         and         (V  -f  i)a  =  o,  imaginary (370 

(c)  Points  on  Y.     Put  x  =  o. 

y  4-  2Ay  —  2y  —  2Ay  -f  y  =  o ; 


-i)  =  o.  >  (37I) 


y  =  o,  origin  ; 

y  —  ±  i,  N.  and  S.  points.  \ (372) 

y  —  —  A  ±  Vi  4-  A2,  P  and  P7,  poles. 

(d)  For  tangents  at  origin, 

y  —  -*"18  =  o,        jj/  =  ^r        and        ^/  =  —  x  tangents (373) 

(<?)  To  find  points  of  the  curve  on  the  line  y  —  x,  put_y  =  x\r\  (364)  and  we  have 

8A*5  —  S*4  -  4Ajtr3  =  o  ; (374) 


x\2 A*'  —  2x  —  A)  =  o  \        _  i  ±  /r+TA3 ; (375) 

(  *  2A~ 

Hence  the  line  y  =  x  cuts  the  curve  once  at  infinity,  Mm-  times  at  origin,  and  at  two  other 


72  AZIMUTH. 

real  points  depending  for  position  on  A.    The  six  points  being  thus  accounted  for,  there  is  no 
inflexion  at  the  origin. 

(/)  To  find  points  of  the  curve  on  the  line  y  =  —  x,  put  /  =  —  x  in  (364),  and  we  obtain 


—  8A.r6  —  8*4  +  4A*3  =  o  ;    .........     (376) 


(377) 


2A  j 

The  same  conclusions  as  in  (e). 

Also  by  taking  cos  Z  equation,  (177),  and  putting  cos  Z  =  ±  V$  (i.e.,  y  =•  x), 


^'' 

we  obtain  sin  h  =  +  I         and         sin  h  =  +  — 


FIG.  s.        Therefore  h  =  ±  90°         and        h  =  sin  - x  /  ± 


. 
l/i 


V 

2  Ay 


For  the  line  j  =  —  x  we  obtain  the  same  value  of  h  as  for  y  =  x,  since  cos  Z  =  ±  t/£ 
corresponds  also  to  j/  =  —  ;tr. 

The  results  in  (e)  and  (/)  indicate  for  the  parts  above  the  horizon  the  form  in  Fig.  8, 
and  inflexion  occurs  at  some  point  K  below  X. 

(g)  To  find  form  of  curve  at  N.  and  S.  points. 

For  N.  put  y  =  y  -\-  \. 

The  coefficient  of  y  is  4A;  always  -f- 
The  coefficient  of  x*  is  (2A  —  4)  ;  -j-  when  A  >  2,  —  when  A  <  2. 

Hence  when  A  >  2  the  form  is  shown  in  Fig.  9;  and  when  A  <  2,  as  seen  in  Fig.  10. 


FIG.  9.  FIG.  10. 

For  latitudes  30°,  45°,  and  60°  the  form  as  in  Fig.  10. 
For  S.  point  put  y  =  y  —  I. 

The  coefficient  of  y  is  —  4A;  always  — . 
The  coefficient  of  x*  is  —  2A  —  4;  always  — . 

/.  the  form  is  always  as  in  Fig.  10. 
(K)  For  form  at  P  and  P'. 

Moving  origin  to   P  and  P',  y  =  y  -f-  b,   in  which  b  is  —  A  -f-  I/A*  -f-  I,  for  P,  and 
-  A  —  I/ A2  —  i  for  P',  we  have,— 

Coefficient  of  y  is  \6P  +  ioA£4  -  8£3  -  6A£'  +  2b\. 
Coefficient  of  x*  is  \V  -f  4A£3  —  4^  —  2A£  —  i }. 

For  latitudes  30°,  45°,  and  60°,  substituting  the  values  A  =  l/f,  i,  and  VJ, 
and  the  corresponding  values  of  b,  we  find  form  at  P  and  at  P7  for  all  these  lati- 
tudes, as  shown  in  Fig.  n. 

FIG.  n. 

(/)  To  determine  where  the  curve  cuts  the  asymptotes. 


AZIMUTH. 
In  (364)  let  y  =  x  —  A,  and  we  obtain 

(4Aa+  6)*4-  (8A3+  I2AX+  (8A4-f  8A'+  2)x*- 


73 


2Aa-  2A)*  -  (A1-  A8)  =  o.     (378) 


When  A  =  V\,  this  equation  has  one  positive  and  one  negative  root,  both  less  numerically 
than  unity,  and  has  no  other  real  roots  ;  when  A  =  i,  one  positive  root  less  than  unity,  one 
root  equal  to  zero,  and  no  other  real  roots  ;  when  A  =  1/J,  there  are  no  real  roots. 
Similarly  for  the  other  asymptote,  substitute 


y  =  —  x  —  A. 


(K)  For  limiting  forms  of  curve. 


If 


and  by  (364), 


L  =  90,        A  =  oo, 


(379) 


x1  -|-  y  =  o,  conjugate  point  at  origin,  zenith; 
x*  -\-  y  =  i,  horizon. 


If 


L  =  o,        A  =  o. 


-  2 


-*'  =  0.    .     .     (380) 


The  curve  is  symmetrical  with  regard  to  both  X  and  Y  ; 

tangents  at  origin,  y  =  ±  x  ; 
asymptotes,  y  =  ±  x. 

The  curve  crosses  asymptotes  at  origin  only,  has  inflexion 
at  origin  and  is  shown  in  the  accompanying  diagram. 
98.  Locus  No.  6.      Time-azimuth,  error  in  d.      Arts.  45  and  72  to  75.     Branches — i.  The 
meridian  for  absolute  min.     2.  The  curve  of  algebraic  max.  and  min.,  giving  numerical  max. 
and  min.;  given  by  equation  (194). 


(a) 
Highest-degree  terms  give 


2.d  Branch. 
2  A 


—  /  +  x1  -  o. 


(381) 


(382) 


whence  two  asymptotes  parallel  to  y  -—  ±  x,  and  there  are  no  parabolic  infinite  branches. 


74  AZIMUTH. 

For  asymptote  parallel  to  y  =  x, 


=-A (383) 

-,  =  * 

For  asymptote  parallel  to  y  =  —  x, 


y  =  —  A  ±  i/Aa  +  i,  poles 


(c)  Tangents  at  origin,  y  =  ±  x. 

For  tangents  at  E.  and  W.,  differentiating  (381), 


x 


whence  y  =  x  —  A,        and        y  =  —  x  —  A  are  asymptotes.      .     „     .     .     (385) 

(b)  Points  of  curve  on  X,  put^  =  o. 

x*  —  x*  =  o  ;         /.  x*  =  o,  origin,  zenith  ; (386) 

x  =  ±  i,E.  and  W.  points (387) 

Points  of  curve  on  Y.     Put  x  =  o ; 

f  +  2 Ay  —  y  =  o;         -'-f  =  o,  zenith  ;  , 


/J4I  A   •**        ___      ^  *»     A     A     *VM/  T 

uy  f\x  —  AX  —  A.t\xy 

— —  —  " : — —  _|_ •  faRni 

dx       4.y  —  2y  -\-  2Kx*  -i-  f^A^  \*=± i  ~      -  A  '  W°SV 


•'•  y  =  ±  -T-  are  tangents  at  E.  and  W, (39°) 

» 

(d)  To  ascertain  the  form  of  the  curve  at  Pand  f,  put  y -\-  b  for_y  and  move  origin  to 
(o,  b\ 

The  coefficient  of  y  becomes  4^'  +  6A£a  —  2b (391) 

The  coefficient  of  x*  becomes  2 Kb  -J-  I .     (392) 

For  point  P,  b  =  4/A'  +  i  —  A, 


and  the 


AZIMUTH. 

coefficient  of  y  is  always  -J-, 
coefficient  of  x*  is  always  -j- ; 


.-.  at  .Pthe  form  is  always  as  in  Fig.  n,  art.  97  (h). 


75 


For  point  P', 


The  coefficient  of  y  is  always  —  . 

The  coefficient  of  x*  is  -(-  when  A  <  |/T  giving  form 

The  coefficient  of  x*  is  -  when  A  >   ^  gjving  form  same  as  at  P. 

0)  For  curve  crossing  its  asymptote.     Combine  first  of  (385)  and  (381). 
Eliminating/,  we  have 


x  = 


t        u-  u 
for  which 


I  +  A'  ±  |/i  -A4 


2A 


(393) 


(/)  Therefore,  when  A  <  I,  asymptote  crosses  curve  in  two  real  points  ; 

when  A  =  i,  asymptote  is  tangent  to  curve  (and  coincides  with  the  tan- 
gent to  the  curve  at  E.  and  W.  points)  ; 

when  A  >  i,  asymptote  does  not  cross  the  curve. 
(g)  Combining  second  of  (385)  and  (381),  we  find 


A2)  ±  VT^"A4 


2A 


_ 
i  -  Aa  ±  Vi  -  A4 


.         , 

(394) 


and  the  same  condition  as  by  (/). 
(h)  For  limiting  forms  of  curve. 


If 


L  =  o,        A  =  o, 


equation  (381)  becomes 


y"  —  x"  —  f-\-x*=o', 


(395) 


(396) 


76  AZIMUTH. 

Two  line^  =  ±x,(Z  =  45°  and  135°),  and  primitive  circle,  horizon. 

(i)  If  L  =  90,        A  =  oo, 

(381)  becomes  2Ay  -(-  2A;r2j  —  o  ; 

j(y  +  *')  =  o; (398) 

whence  axis  of  x  and  conjugate  point  (zenith). 

99.  Locus  No.  7.  Time-altitude-azimuth,  error  in  h.  Arts.  46  and  76  to  79.  Branches 
• — I.  The  prime-vertical  for  absolute  max.  2.  The  meridian  for  absolute  min.  3.  The  hori- 
zon for  absolute  min.  4.  For  algebraic  max.  and  min.,  giving  a  numerical  max.  by  equation 
(207). 

The  first  three  branches  are  defined. 


Branch. 
By  equation  (207), 


*y  +  2  Ay 

-  4A*y  -  2AS  +/  =  o.     (399) 

(a)  Highest-degree  terms,  y*  (y*  -f-  2^y  -f-^"4)  ;     ..........     (400) 


gives  no  real  infinite  branches. 
The  terms  2Ajr"  -j-^V4,  put  equal  to  zero,  indicate  a  parabolic  branch, 


(40i) 

(b]  To  determine  contact  of  curve  with  axes.     Points  on  X. 
Put  y  =  o ; 

2Ay  —  2A^r4  =  o;        .•.  x"  (x*  —  i)  =  o ; (402) 

Whence  x"  =  o,  ze  nith  ; (403) 

x—  ±  i,  E.  and  W (404) 

Points  on  Y.     Put  x  =  o ; 

y  +  2  Ay  —  2y  —  2  Ay  -i-y  =  o; 


AZIMUTH.  77 

=  o,  zenith ;                                     \ 
y  =  ±  i,  N.  and  S. ;  >- (406) 

y  =  -  A  ±  Vi+A\  P  and  P'.      ) 

/_....     (407) 


For  form  at  origin  y*  =  2Ajr4.     .     .     . 

(c)  For  tangents  at  E.  and  W. 


dy-\         - 

--      =  -  ----  =  ±  A;  ........     (408) 


j/  =  ±  A^r,  tangents  at  E.  and  W  .........     (4°9) 

(d~)  To  determine  form  of  curve  at  N.  and  S. 
Put  y  —  y  -\-  i,  and  move  origin  to  N. 

Coefficient  of  y  is  4A,    always  -}-. 

Coefficient  of  x?  is  2A  —  4  ;  —  if  A  <  2  ;  -f  if  A  >  2. 
Hence,  when  A  <  2 


When  A  >  2 

From  next  higher  terms  when  A  =  2,  same  as  A  >  2. 

Put  y  —  i  for  jj/,  and  move  origin  to  S. 

Coefficient  of  y  is  —  4A. 

Coefficient  of  x*  is  2A  +  45         •'•  at  S.  always  as  A  <  2  for  N. 

(e)  To  determine  the  form  of  curve  at  P  and,  P'. 
Put  y  -(-  b  for  y,  and  move  origin  to  (o.#). 

The  sign  of  the  coefficient  of  y  is  found  then  to  depend  on 

3|, (410) 


7  8  AZIMUTH. 

and  the  sign  of  the  coefficient  of  X*  to  depend  on 


(411) 


Substituting  the  values  of  A  and  of  b  for  the  cases  of  the  curves  drawn,  lats.  30°,  45°,  60°, 
we  find  from  the  signs  of  (410)  and  (411)  the  form  of  the  curve  at  P  and  at  P'  to  be  the  same 
as  at  N.  for  A  >  2. 

The  values  of  A  and  b  being  as  follows  : 


For  P,  since  b  =  —  A  +  4/Aa  +  i, 

Lat.  30°;  A  =  l/£,  b  =  Vl>\ 

Lat.  45°;  A=  i,  b  =  V~2  —  I  ; 

Lat.  60°  ;  A  =  4/J,  b  =2  —  V$. 

For  P',  since  b  =  —  A  —  ^A'  -f  i, 

Lat.  30°  ;  A  =  VI  b  =  -  V$- 

Lat.  45°  ;  A  =  I,  b  =  —  \/2  —  i  ; 

Lat.  60°  ;  A  =  i/J,  b  =  -  i/J  -  2. 

(/)  To  find  circles  of  altitude  tangent  to  locus. 
Let  *'+y  =  c'  .............     (412) 

Substitute  in  (399)  X*  =  c*  —y*  to  eliminate  x,  and  we  have  a  cubic  in  y  : 

y  (i  +  <7-4X-  2^(1-0  =  0;    .......  (413) 


~  -  /       x 

Or>  /-(Tqr?7^-  -<n^r  °  .......  (4I4) 

The  condition  for  equal  roots  is 

AV  (i  -  g1)*  64^" 


AZIMUTH. 


79 


which  reduces  to  A'  (i  —  r'Y  =  -  ,   4.g    ...  .  (416) 

27(1  -|-  c  ) 

Clearing  and  arranging  in  the  form  of  a  quadratic  in  c, 

°  +  644 

T£-' 

2?A'  +  32  ±  8 
solving,  **=-  -^A"  -    ........    (4i8) 

which  always  gives  two  real  positive  values  of  c*  and  thence  always  two  positive  values  of  c. 
(£•)  To  find  limiting  forms  of  locus, 

L  —  90,        A  =  oo, 
in  (399)  2/  +  6*y  +  6*y  -f  2x*  —  2/  -  44r>a  -  2x"  =  o  ;     .     .     .     .     (419) 

i)  =  o;  .........     (420) 

whence  peculiar  point  (/  +  *8)  =  o,  zenith  ;      ..........    (421) 

x3  +y  =  I,  circle,  horizon  ..........     (422) 

(K)  L  =  o,        A  =  o, 

in  (399)  /  +  2^y  +  ^y-2y-6^y-44r>+/  =  o 

y  +  2^>4  +  jry  -  2j/4  -  6^y  -  4**  +/  =  a 

Highest-degree  terms  y1  (y  +  ^a)s  =  o,  gives  infinite  branches  whose  asymptotes,  parallel 
to  X,  by  coefficient  of  highest  power  of  x  are 

y-4  =  o,        y  =  ±2  ...........     (424) 

For  form  at  origin  y1  =       4%*  ;  i 

y  =  ±  ™*>  -  ~V^-  ......   •   (425) 

(*)  For  contact  with  X  and  Y. 

4  =  o,  at  origin  only,  zenith  .........     (426) 


y_2y+y=0; 


(427) 


8o 


AZIMUTH. 


y  = 


=        o,  origin 
±  i,  double 


1 o        .         .         .         (428) 

points.  ) 


(/£)  For  form  at  (o,  ±  i)  move  origin,  and  lowest-degree  terms  give 


tangents, 


x*  =  o ;      ) 

=  ±x.      \ 


(429) 


(/)  Combining  (424)  with  (423),  we  find  that  the  curve  does  not  cross  its  asymptotes. 


Form  when  A  =  o. 


100.  Locus  No.  8.  Time-altitude- azimuth,  error  in  t.  Arts.  46  and  80  to  83.  Branches 
— I.  Absolute  max.,  the  prime-vertical.  2.  Absolute  min.,  the  six-hour  circle.  3.  Algebraic 
max.  and  min.,  giving  numerical  max.  and  min.,  the  meridian.  We  have,  also,  the  equator 
for  a  branch  giving  a  constant  error. 

Though  all  branches  are  great  circles,  and  therefore  directly  defined  in  the  projection, 
yet  it  may  be  of  interest  to  analyze  the  equation  to  the  equator  ;  that  of  the  six-hour  circle 
having  been  treated  in  art.  94. 

The  equation  to  the  equator,  by  (223),  is 


(430) 


From  y  -f-  x*,  there  is  no  infinite  branch. 


Contact  with  X,  put 


Contact  with  Y,  put 


=  ±  i,  E.  and  W. 


(431) 


•'•y  = 


A 


(432) 


AZIMUTH.  8  1 

For  tangents  at  E.  and  W.,  move  origin  to  (±  i,  o), 

y  =  ^f  hx,  tangents  at  E.  and  W.     .     .     ......     (433) 

Limiting  forms, 

L  =  o,  A  =  o,  y  =  o,  axis  of  X,  the  prime-vertical  ; 

L  =  90°,         A  =   oo,        y*  -\-  x*  =  i,  primitive  circle,  horizon. 

/ 

101.  Locus  No.  9.     Time-altitude-azimuth,  error  in  d,    Arts.  46  and  84.  Branches  —  i.  The 
meridian  for  absolute  min.     2.  The  prime-vertical  for  absolute  max.     3.  The  curve  of  elonga- 
tions, q  =  90°,  for  algebraic  max.  and  min.  giving  numerical  min. 

The  third  branch  has  been  treated  in  art.  58.  It  is  noteworthy  that  this  curve,  q  =  90°, 
though  it  has  occurred  frequently  in  the  loci  preceding  No.  9,  yet  in  no  case  heretofore  giveh 
has  it  been  the  locus  of  algebraic  max.  and  min. 

102.  Locus  No,  10.     Time-azimuth  and  altitude-azimuth,  for  error  in  L,  giving  the  same 
numerical  error  in  the  computed  azimuth.     Arts.  85  to  90.     Branches  —  i.  Errors  equal,  hav- 
ing signs  alike,  the  prime-  vertical  and  the  curve  of  q  =  90.     2.  Errors  equal  numerically  with 
contrary  signs. 

id  Branch. 
By  equation  (253), 

/  +  3*y  +  2*4  +  2  A/  -f  2hx*y  -  /  -  2*'  =  o  ......     (434) 

(a)  Highest-degree  terms  have  no  real  factors;  therefore,  there  is  no  infinite  branch.    No 
real  factors  in  terms  of  lowest  degree  ;  hence,  no  branches  through  the  origin. 

(b)  For  contact  with  X  and  Y. 


Put  j  =  o,        2*4  -  2*'  =  o,        **(**—  i)  =  o;    ......     (435) 

x*  =  o,  conjugate  point  at  origin,  zenith;  ) 
*=±i,E.andW.  f       ......     (436) 

Put  x  =  o,        y  +  2A/-/  =  o,         or        ?(f  +  2hy-  i)  =  o;  .     .     .     (437) 

y  =  O,  conjugate  point  at  origin  ;        y  =  —  A  ±  Vi  -\-  A",  P  and  P7.  .     .     (438) 

(c)  To  find  form  of  curve  at  E.  and  W. 

Moving  origin  to  (i,  o)  in  (434).     Terms  of  lowest  degree, 


82  AZIMUTH. 


—  2.X  ,  _ 

give  y  =  — x —  for  tangent  at  E. ; 

2.X 

similarly  y  =  -r-  for  tangent  at  W. 


y 

Numerical  values  of  —  for  different  latitudes: 
x 


(439) 


3.47  in  lat.  30°,         2  in  lat.  45°,         1.15  in  lat.  60° (440) 


(</)  For  forms  at  P  and  P'. 

For  P,  move  origin  by  substituting 


y  —  A  +  VA3-}-  i  iory. 


The  coefficient  of/  is  found  to  be  2  (A*  -f-  i  —  2A  V  A2  -f-  i)  4/A"  -f-  I,  always  -\-. 


The  coefficient  of  X?  is  found  to  be  4  A"  -|-  i  —  4A  I/A3  -f-  i, 

which  is  -f-  when  A  <  \  V~2  and  —  when  A  >  \  V2. 

For  the  curves  drawn  for  lat.  30°,  45°,  and  60°,  the  smallest  value  of  A  is  £  V~$  (lat.  30°), 
which  is  greater  than  J  ^2 ;  hence,  for  the  latitudes  used,  the  coefficient  of  3?  is  in  all  cases 

minus  and  the  curve  at  P  is  of  the  form 


For  P',  substitute  y  —  A  —  VA*  +  I  for  y,  and  we  find 

the  coefficient  of  y  to  be  always  —  , 
and  the  coefficient  of  x*  to  be  always  -f-  ; 

hence,  form  at  P'  always  as  shown  for  P  above. 
(e)  For  limiting  forms  of  curve. 

If  L  =  o,         A  =  o, 

equation  (434)  becomes  f  -f  $x*y*  -f  2;tr4  -  /  —  2x*  =  o, 


=  o,  ) 

=  o.  i 


+  ^  =  i,  .primitive  circle,  horizon;  | 

-[-  2.T1)  —  o,  conjugate  point  at  origin,  zenith,  f  ' 


If  L  =  90°,         A  =  oo, 


AZIMUTH.  83 

the  equation  becomes       2y3  -}-  2x*y  =  o,         or        y  (}?  -J-  x*}  =  o  ;    .     =     .     .     .     .     .     (443) 

y  =  o,  giving  X ; 
jj/a  -f-  x*  =  o,  conjugate  point  at  origin. 


103.  Though  this  treatise  is  called  Azimuth,  and  though  an  attempt  has  been  made  to 
present  the  subject  in  a  new  light  on  some  points,  no  attempt  has  been  made  to  put  into  the 
hands  of  the  worker  in  the  field  the  practical  methods  of  observation  for  the  elimination  of 
errors.     The  Coast  Survey  office,  in  its  appendices*  to  the  reports,   leaves  nothing  to  be  de- 
sired on  the  score  of  precision  in  work  with  refined  instruments.     For  the  navigator,  provided 
with  the  sextant,  there  is,  it  is  hoped,  something  gained  by  the  investigations  pursued,  appli- 
cable as  well  to  more  refined  work.     Even  with  the  sextant,  when  the  observation  on  shore 
for  determining  the  direction  of  the  meridian  cannot  be  taken  at  the  most  favorable  instant, 
errors  in  the  data  may  sometimes  be  eliminated  by  observing  at  the  same  relative  points  both 
east  and  west  of  the  meridian.     Even  though  the  error -by  the  observation  on  one  side  of  the 
meridian  may  be  large,  the  mean  of  the  results  on  both  sides  may  leave  no  resulting  error  in 
the  computed  azimuth. 

But  the  azimuth  problem  has  also  been  particularly  useful  in  studying  the  variations  of  the 
astronomical  triangle.  For  this,  the  problems  on  time  and  on  latitude  do  not  bring  out  so 
many  truths.  Inspection  of  the  terms  in  which  the  errors  are  expressed  usually  suffices  in 
the  two  problems  mentioned.  Not  so,  often,  in  the  azimuth  problem,  though  much  will  be 
obvious.  .,, 

104.  A  single  instance  will  suffice.     Taking  the  case  dZ  —  -      '• -j —  dt  (58)  for  error 

in  Z,  due  to  a  positive  error  in  the  hour  angle,  which,  by  art.  34,  employing  the  general  astro- 
nomical triangle,  is  error  in  time.  Having  regard  to  the  signs  of  trigonometric  functions  and 
to  the  resulting  sign  before  the  whole  expression  necessitates  a  knowledge  of  the  approxi- 
mate values  of  the  particular  parts  of  the  triangle  found  in  the  expression  within  the  limits  of 
quadrants.  We  look  to  the  resulting  sign  of  dZ  to  determine  whether  it  is  an  algebraic  max. 
or  min.  at  particular  points.  If  -j-  and  dZ  changes  from  an  increasing  function  to  a  decreas- 
ing function,  we  shall  have  an  algebraic  max.  If  -|-,  and  the  change  is  from  a  decreasing 
function  to  an  increasing  function,  we  shall  have  an  algebraic  min.  If  —  and  dZ  changes 
from  a  numerically  increasing  function  to  a  decreasing  function,  we  shall  have  an  algebraic 
min.,  though  a  numerical  maximum  negative.  If  —  and  change  is  from  a  numerically  de- 
creasing function  to  an  increasing  function,  we  shall  have  an  algebraic  max.,  but  a  numerical 
min. 

From  (58),  by  inspection  alone,  we  cannot  tell  whether  there  is  any  point,  other  than  on 
the  meridian,  where  the  function  changes  from  increasing  to  decreasing  or  the  contrary  ;  but 
if  there  be  such  a  point  for  -|-  d  <  L,  for  instance,  we  know  from  the  change  that  occurs  on 
the  meridian  at  upper  culmination  that  here  will  be  an  algebraic  min.  (numerical  max.  neg.)  ; 
and  at  the  point  of  change  before  lower  culmination  we  shall  have  an  algebraic  max.  (nu- 
merical min.  negative)  ;  at  lower  culmination  an  algebraic  min.  (numerical  max.  negative),  not 
having,  however,  the  same  numerical  value  as  at  upper  transit,  the  latter  being  greater;  at 
the  next  point  east  of  the  meridian  an  algebraic  max.  (numerical  min.  negative).  The  curve 

*  By  Charles  A.  Schott. 


84  AZIMUTH. 

passing  through  W,  Z,  E,  as  found  in  No.  4,  determines  for  a  given  declination  whether  the 
star  reaches  any  point  on  it ;  then  we  can  discriminate  between  the  algebraic  max.  and  min. 
If  there  is  a  point,  we  have  already  seen  how  to  discriminate.  If  there  is  no  point  on  the 
curve,  as  for  certain  declinations,  when  the  observer  is  in  very  high  latitudes,  then  we  have 
the  meridian  alone  to  consider  ;  d  being  less  than  Z,  at  upper  culmination  we  shall  have  an 
algebraic  min.  (numerical  max.  negative),  and  at  lower  culmination  an  algebraic  max.  (numeri- 
cal min.  negative),  for  cos  h  will  be  greater  for  lower  transit ;  hence  dZ  numerically  less  but 
negative. 

105.  In  respect  to  the  elimination  of  errors  by  observing  at  symmetrically  situated  points 
on  both  sides  of  the  meridian,  notwithstanding  a  poor  point  of  observation  for  one  side 
alone: — In  No.  I,  No.  2,  and  No.  3,  if  the  body  is  observed  at  the  same  altitude  on  both 
sides  of  the  meridian,  the  error  due  to  error  in  h,  that  due  to  error  in  Z,  and  that  due  to 
error  in  a?  will  be  eliminated.  Therefore,  since  in  No.  I,  even  though  the  error  in  h  should 
not  be  eliminated,  yet  if  the  star  is  observed  at  q  =  90°  the  error  will  reduce  to  zero,  we 
should  endeavor  to  observe  at  that  point  on  both  sides  of  the  meridian  ;  therefore  select  a 
close  circumpolar  star.  But  if  the  star  crosses  the  prime-vertical  we  should  observe  at  its 
crossing  the  curve  of  min.  errors  for  error  in  h  on  both  sides  of  the  meridian. 

In  No.  4  the  error  in  t  cannot  be  eliminated  by  the  mean  of  the  results,  but  if  q  =  90° 
for  d  >  L,  it  will  be  zero,  and  if  d  <  L  there  is  a  point  of  min.  error ;  this  point,  then,  or 
q  =  90°,  should  be  selected  on  both  sides  of  the  meridian,  for  then,  by  No.  5  and  No.  6,  the 
errors  in  L  and  d  will  be  eliminated.  Therefore  the  value  of  a  close  circumpolar  star  to  apply 
these  conditions. 


APPENDIX. 


A  SCHEME  FOR  DERIVING  CURVES  FROM  THE  VARIATIONS  OF  THE  SPHERICAL  TRIANGLE. 

I.  The  curves  already  obtained  from  the  problems  on  azimuth  suggest  the  possibility 
of  obtaining  a  great  variety  of  loci  derived  from  the  variations  of  the  spherical  triangle  ;•  and, 
from  the  astronomical  triangle,  by  regarding  always  L  and  d  constant  in  finding  the  maximum 
and  minimum  effects  of  variation  in  any  part,  due  to  the  variation  in  any  other  part,  we  may 
find  many  interesting  curves,  though  the  number  will  be  less  than  if  not  restricted  to  L  and  d 
constant. 

2.  The   ultimate  equation  in  x  and/  may  be  disengaged  or  not,  at  one's  pleasure,  from 
any  idea  of  the  sphere,  since  A  (or  B,  or  C)  will  be  a  single  arbitrary  constant  (see  art.  55)  ; 
but  the  sphere  furnishes  the  astronomical  triangle  as  the  means  of  obtaining  the  equations. 
Therefore  there  may  be  curves  of  interest  to  the  mathematician,  though  having  no  utility. 

3.  To   adopt   a  system  for  obtaining  the  equations  to  the  loci,  we  have  the  following : 
1st.  Any  four  parts  of  the  triangle  involved,  and  any  three  of  these  given,  the  remaining 

part  may  be  found. 

2d.  Form  all  possible  groups  of  four  parts. 

3d.  In  each  group,  each  part  may  be  taken  in  turn  as  the  one  to  find. 

4th.  In  every  case,  the  partial  differential  coefficient,  for  each  given  part  in  error  and  the 
part  to  be  found,  may  be  taken. 

5th.  Each  partial  differential  coefficient*  will  have  some  kind  of  max.  and  min.,  as  defined 
for  the  purposes  of  this  treatise  ;  assuming  always,  in  determining  these  max.  and  min.,  that 
L  and  d  are  constant,  in  order  to  conform  to  the  nature  of  the  case  in  the  problems  used  in 
practice,  even  though  L  or  d,  or  both,  may  have  been  variable  in  the  problem  giving  the 
partial  differential. 

6th.  If  algebraic  max.  and  min.  occur,  the  differential  coefficient's  own  first  differential  put 
equal  to  zero  will  show  their  existence  from  the  equations  derived,  all  parts  except  L  and  d 
varying. 

7th.  If  there  were  an  instrument  to  measure  q  accurately,  and  if  our  instrument  for 
measuring  2T gives  precision,  and  if,  in  nature,  there  were  problems  demanding  Z  and  q  as 
given  parts,  then  all  possible  problems  based  on  the  astronomical  triangle  might  be  of  utility. 

8th.  Lacking  utility,  all  possible  cases  will  still  give  examples  in  curve-tracing. 

9th.  Great  circles  of  the  sphere  will  probably  recur  many  times  in  the  loci  of  the  preced- 
ing article,  as  they  have  done  in  the  curves  Nos.  I  to  10  defined  in  this  treatise  ;  and  possibly 
the  same  novel  loci  that  those  curves  give,  or  that  may  be  found  in  the  future,  will  also  recur. 

loth.  Making  the  combinations,  we  have  90  individual  cases;  each  one  having  a  recipro- 
cal, which  need  not  be  considered,  thus  making  180. 

*  Excepting  when  it  consists  of  functions  of  the  constanis.  only. 


86  APPENDIX. 

nth.  In  these  ninety  cases  are  included  the  equations  to  curves  found  in  this  treatise, 
and  equations  for  additional  useful  problems  (time  ;  latitude  ;  computation  of  the  altitude 
in  the  problem  of  astronomical  bearing;  and  the  time-altitude-latitude-azimuth,  in  distinction 
from  time-altitude-declination-azimuth,  called  in  this  work  time-altitude-azimuth). 

I2th.  In  making  the  cases  for  partial  differentials,  the  two  constants  will,  of  course,  be 
given  parts  in  the  original  problem,  and  either  one  of  the  variables  will  be  another  given 
part ;  the  remaining  variable,  a  part  to  be  found. 

1 3th.  For  illustration  : 


1st  A,  b,  and  c,  being  given,  to  find  B.    A  and  b  constant,  c  and  B  variable,  we  shall  indi- 
dB^  dB_ 
dc  '  dc. 


cate  by  writing  A,  b,  -j—\  7-  being  the  coefficient  of  error  in  B  due  to  a  small  error  in  c.     For 


max.  and  min.  effects,  we  have  d  \~~r~]  =  O»  regarding  b  and  c  (corresponding  to  co-Z-  and  co- 

\d£  * 

din  the  companion  triangle),  always  as  constant  in  this  part  of  the  work. 
2d.  For  the  reciprocal : 

dc 

Given  A,  b,  and  B,  to  find  c.     A  and  b  constant,  B  and  c  variable,  A,  b,  -T~,  error  in  c,  due 

to  error  in  B. 

The  reciprocal  will,  in  all  cases,  be  omitted  as  giving  the  same  locus  as  the  first  form,  but 
having  max.  and  min.  interchanged  ;  and  to  simplify  the  indication  we  shall  omit  the  symbol 

.   B 
a,  writing  only  A,  0,  —. 

C 

I4th.  For  combinations  of  four  of  the  six  parts — A,  B,  C,  a,  b,  c — of  the  triangle  we  have 

1.  A,£,C,a (6) 

2.  A,  B,  C,  b (7) 

3«  A,B,  C,  c (8) 

i 

4.  A,  £,  a,  &. (3) 

5-  A,B,a,c (4) 

6.  A,B,b,c (2) 

7-  A,  C,a,& (9) 

8.  A,  C,  a,  c (10) 

9.  A,  C,  &,c (11) 

10.  A,  a,  b,  c (5) 

11.  £,C,a,6....- (12) 

12.  B,  C,  a,c (13) 

13-  B,C,b,c (14) 

14.  B,  a,  b,  c (i) 

15.  C,  a,  b,  c. (15) 


APPENDIX.  87 

The  numbers  (i),  (2),  and  (3)  show  the  parts  taken  for  the  curves  described  in  this  treat- 
ise. (4),  the  parts  for  the  time-altitude-latitude-azimutlt ;  and  (5)  those  for  the  time-sight,  the 
problem  of  latitude  at  any  time,  and  the  computation  of  h  for  use  in  the  astronomical  bearing. 

1 5th.  From  the  astronomical  triangle, 

A  =  t,        B  —  Z,         C=q,        a  =  co-h,        b  =  co-d,        c  =  co-L  ; 
we  derive  the  following  cases  for  loci : 

(1)  Parts  involved,  £,  a,  b,  c;  Z,  h,  d,  L. 

D  ^ 

a,  b,  — ;   h,  d,  -j  \  alt.-az.,  error  in  L (a) 

C  J_^ 

B  Z 

a,  c,  T  ',  h,  L,  -j  :  alt.-az.,  error  in  d. (b) 

B  Z 

b,  c,  —',  h,  L,-r  :  alt.-az.,  error  in  h .     .     .     (c) 

Ct  rL 

h  // 

B,a,-^;Z,h,-£ (d) 

B,b,-;Z,dyj^ (e) 

a              h 
B,  c,  -T',  Z, L,—, (/ ) 

(2)  Parts  involved,  A,  B,  b,  c\  t,  Z,  d,  L. 

B  Z 

A,  b,  —  ;    t,  d,j:  time-az.,  error  in  Z- (g) 

C  JLs 

X" 

B  Z 

A,  c,-,;  t  L,  -j  :  time  az.,  error  in  d. (h) 

b  a 

B  Z 

b,  c,  -.;d,L,-\  time-az.,  error  in  / (i) 

A  t 

B,b,-;  Z,d,j. (*) 

c  L, 


88  APPENDIX. 

(3)  Parts  involved  A,  B,  a,  b ;  /,  Z,  h,  d. 


B  Z 

A,  a,  -T  ;  t,  h,  —, :  time-alt.-dec.-az.,  error  in  d. (») 

B  Z 

A,  b,  — ;  t,d,j-:  time-alt.-dec.-az.,  error  in  h (o) 

,  B  Z 

a,  b,  -;•,  A,  a,-:  time-alt.-dec.-az.,  error  in  t (fi) 

*i  T 


A,B,ar,    t,Zhj fr) 

A  t_ 

B,b,  -;    Z,d,-H        (S) 


(4)  Parts  involved,  A,  J3,  a,  c\  ty  Z,  h,  L. 


B  Z 

A,  a,  —;  t,  h,  j  :  time-alt.-lat.-az.,  error  in  L. 

T>  rp 

A,  c,  —  ;  t,  L,  j  :  time-alt.-lat.-az.,  error  in  A. 

T>  ^ 

a,  c,  -7;  h,  L,  —  :  time-alt.-lat.-az.,  error  in  t. 

jfi  t 


r>  rr    i  ( »,\ 

B,a,-;  Z,  h,  -L 

B,f,^;Z,L,^..        « 

a  h 


(5)  Parts  involved,  A,  a,  b,  c;  /,  h,  d,  L. 

A  t  (  time-sight,  error  in  L.                                 j 

c  '  L  (  reciprocal,  for  lat.  problem,  error  in  t.      r  •    •    -    •  (A) 

A  .    t 

a,  c,  —  ;  A,  Z,,  —  :  time-sight,  error  in  d. n$\ 

,       A  t  (  time-sight,  error  in  h. 

b,  c,  —  •  a,  L,  -7 :  •{ 

a  n  (  recip.,  alt.  computed,  error  in  t. 

c  L 

A,  a,  -r  ;  /,  h,  -, :  lat.  problem,  error  in  d. (D) 

.    ,   c  L  (  lat.  problem,  error  in  Ji.                          )                          ,  „ 

'  a  '  '  h  '  \  recip.,  alt.  computed,  error  in  L,           } 

a  r   h 

A,  c,  T  ;  t,  L,  =  :  alt.  computed,  error  in  d.       •     • 


APPENDIX. 
(6)  Parts  involved,  A,£,  C,  a;  t,  Z,  q,  h. 


*9 


a  n 

A    CB  ,Z- 

A,  (-,  -  ,     t,  g,  fi. 

B          ,   Z 

A          >V        •  /         M         

A,  a,  c,    *>**-• 

B  C^-  Z  a*- 

B>  6'  a  '  *  9'  h' 

A          ,   t  ,,, 

B,  a,  -~;  Z,  k,-. (^) 

s*        A          r   *  (M\ 

I      SI        —    •      /7     h     —          .......•••••  \LVJ.  ) 

Ixj  t*,        TJ    >      ff)    ft-t    *y  »         / 

-D  ^ 

(7)  Parts  involved,  A,  £,  C,  &;  /,  Z,  ^,  </. 

r^     ^  ^ 

A,B,j-\t,Z,-j. (N) 

0  M> 

-  ^6'f;  '•*!• w 

5  Z 

A,b,-£\t,d,- (P) 

B,C,j\Z,i9j (Q) 

B,b,^',Z,q,- (R) 

A  t 

C,  b,   -^  ;  q,  d,  ^ 

A?  ^ 

(8)  Parts  involved,  A,B,C,c\  t,  Z,  ?,  Z. 

^^  ?.  /z?:  

-«,^, -;  r,^,r 

A,C,-',    t,g,j. (^ 

c  A 

A  c     -•  t  LZ-  -(V) 

**}  ct       S"  i  /7  " 

B  C  *'  Z  a-  •     '     '     '(W) 

Js,    \s i  i    £">    Y<     7* • 

^r  -t> 

^4               t 
B,c,    -£-,  Z,L,-. 

A  t 

C,c,    B\  q,  L,  g. 


90  APPENDIX. 

(9)  Parts  involved,  A,  C,  a,  b\  /,  q,  h,  d. 

„  a  h 


a 

A 


(10)  Parts  involved,  A,  C,  a,  c,  ;  t,  q,  h,  L. 


(n)  Parts  involved,  A,  C,  b,  c\  t,  q,  d,  L. 

.   b  d 


C 


(a) 


C              q                  % 
A'  "'  ~b  '    tj  k'  d (b) 


(e) 
(0 


a  h 

A'  C'  c  ;  *'  ?>  L ® 

C  Q 

A,  a,  -•  f,  h,  \ (h) 


/-* 

-;  AA| .  (o 

^;    g.hSj. (k) 

-•     L*  m 

a  '  9'  L'  h' 

A  ,  t 


(m) 


(n) 


A     L      C  J     9 

A,  b,  -  ;    /,  d,  ^ (o) 

C               q 
A,c,  j-;  t,L,j (p) 

A              t 
I*,  0,   =  ;    q,  d,  -j (q) 


(s) 


APPENDIX.  91 

(12)  Parts  involved,  B,  C,  a,  b;  Z,  q,  h,  d. 

a 


B,C, 

V 

Z,q,'d.     .     .  ..,.  -,_.     .     .     .     . 

.     .     .     (t) 

B,  a, 

c 

7  ,  q 

.     .     .    (u) 

c 

q 

B,b, 

z'd>'h-   

*   •     •     (v) 

C,a, 

B 

b' 

Z 

.     .     .    (w) 

B 

z 

C,b, 

a  ' 

.     .     .    (x) 

a,  d, 

B 

Z 

.   .  •  (y) 

(13)  Parts  involved,  B,  C,  a,  c;  Z,  q,  h, 

L. 

- 

^.                                               B,C, 

-  ; 

**i-  •  •.  

...   (A) 

B,a, 

c  ' 

Z'h'  L'     '     '     '     '     

.    .    .    (B) 

'       '          .                          .        B,c, 

a  '' 

Z  Lq 
^^  h' 

.    .     .    (C) 

C,a, 

B 

7; 

q'h'L     

.     .    .   (D) 

C,  c, 

B 

Z 

.    .     .  (E) 

a,  c, 

C' 

Z 

'    V 

.    .    .    (F) 

(14)  Parts  involved,  B,  C,  b,  c  ;  Z,  q,  d, 

L. 

B  C 

b 

d 

.    .    .    (G) 

' 

c 

',  y,  £ 

B,b, 

C 

„    ,  q 

.    .     .  (H) 

6 

** 

B  c 

c 

Z  Lq 

...     (I) 

b' 

'    '  d' 

C,b, 

B 

qdZ                                        -     •     • 

.    .    .  (K) 

t. 

** 

C,  c, 

B 

b'' 

Z 

d 

.    .     .    (L) 

b,  c, 

?. 

d,  L,     

.    .    .  (M) 

-^  ,    t*,  J-r, 

6  q 


92  APPENDIX. 


(15)  Parts  involved,  C,  a,  b,  c  \  q,  h,  d,  L. 


b  d 

C,a,-;q,k,7  .............   (N) 

C  J-* 

C,btl;    q.dfj.     ............   (O) 

a  r   h 

C,  c,  -£  ;  g,  L,  j  .............   (P) 

>-^ 

a,b,  -;    A,  4  |  .............   (Q) 

• 

x-« 

a,  c,    -;  h,  L,     .............  (R) 


(S) 


4.  Fundamental  equations  from  trigonometry  for  the  solution  of  the  various  problems 
of  finding  any  part  of  the  triangle,  three  parts  being  given  ;  needed  for  transforming,  in  find- 
ing equations  to  the  curve  of  max.  and  min.  variations,  in  any  three  terms  ;  and  used,  if  de- 
sired, for  differentiating  directly  for  the  general  expression  of  error  in  one  part  due  to  an 
error  in  some  other  part.  For  convenience,  these  equations  are  written  here,  and  the  parts 
involved  written  at  the  left  hand  ;  (i),  (2),  etc.,  conforming  to  the  groups  given  in  the  I4th 
and  1  5th  sections  of  art.  3. 

B,  a,  b,  c.       cos  b  =  cos  c  cos  a  -f-  sin  c  sin  a  cos  B  ........     (i) 

A,  £,  b,  c.  sin  A  cot  B  =  sin  c  cot  b  —  cos  c  cos  A  ........     (2) 

A,B,a,b.  sin  a  sin  £  =  sin  b  sin  A  .............     (3) 

A,  B,  a,  c.  sin  B  cot  A  =  sin  c  cot  a  —  cos  c  cos  B  ........     (4) 

A,  a,  b,  c.  cos  a  =  cos  c  cos  b-\-  sin  c  sin  b  cos  A  ........     (5) 

A,  B,  C,  a.  cos  A  =  —  cos  B  cos  C  -\-  sin  B  sin  C  cos  a  ......     (6) 

A,  B,  C,  b.  cos  B  =  —  cos  C  cos  A  -j-  sin  C  sin  A  cos  b  ......     (7) 

A,  B,  C,  c.  cos  C  =  —  cos  A  cos  B  -f-  sin  A  sin  ^  cos  £  ......     (8) 

A,  C,  a,  b.  sin  C  cot  ^  =  sin  b  cot  #  —  cos  b  cos  £7.  .......     (9) 

A,  C,  a,  c.  sin  £  sin  A  =  sin  #  sin  C.   ......     ......  (10) 

A,  C,  b,  c.  sin  A  cot  C=  sin  ^  cot  c  —  cos  £  cos  A  ........  (i  i) 

B,  (7,  a,  b.  sin  f  cot  B  —  sin  <z  cot  b  —  cos  a  cos  C.  .......  (12) 

./?,  C,  a,  c.  sin  Z>  cot  C  =  sin  0  cot  c  —  cos  #  cos  B  ........  (13) 

B,  C,  b,  c.       sin  b  sin  C  =  sin  c  sin  B  .............  (14) 

C,  a,  b,  c.       cos  c  =  cos  a  cos  £-f-  sin  a  sin  ^  cos  C.     .......  (15) 


APPENDIX.  93 

5.  Additional  formulas  involving  five  parts,  useful  in  transforming. 

A,  B,  a,  b,  c.       sin  a  cos  B  —  sin  c  cos  b  —  cos  c  sin  b  cos  A (16) 

B,  C,  a,  b,  c.        sin  b  cos  C  =  sin  a  cos  c  —  cos  a  sin  c  cos  Z? (17) 

^[,  £7,  #,  b,  c.       sin  c  cos  A  =  sin  #  cos  a  —  cos  £  sin  a  cos  £.....  (i 8) 

A,C,a,b,c.       sin  #  cos  C  =  sin  b  cos  £  —  cos  b  sin  £  cos  y2 (19) 

A,  B,  a,  &,  c.       sin  #  cos  yi  =  sin  c  cos  #  —  cos  c  sin  #  cos  B (20) 

.#,  C  a,  b,  c.        sin  ^  cos  B  =  sin  #  cos  b  —  cos  «  sin  b  cos  £....-.  (21) 

A,  B,  C,  a,  b.      sin  A  cos  #  =  sin  C  cos  Z?  -j-  cos  C  sin  Z?  cos  « (22) 

^4,  Z?,  (7,  b,  c.      sin  Z?  cos  c  =  sin  ^  cos  C '  -\-  cos  y2  sin  C  cos  £ (23) 

A,  B,  C,  a,  c.      sin  C  cos  #  =  sin  B  cos  y£  -j-  cos  B  sin  y4  cos  c.       ...  (24) 

y4,  /?,  C,  a,  c.      sin  ^4  cos  c  =  sin  Z?  cos  £7+  cos  Z?  sin  C  cos  «.        ...  (25) 

A,  B,  C,  a,  b.     sin  B  cos  #  =  sin  C  cos  ^4  -f-  cos  C  sin  ^4  cos  b (26) 

^4,  ^,  £",  b,  c.      sin  C  cos  b  =  sin  ^4  cos  B  -(-  cos  ^4  sin  B  cos  c (27) 

6.  The  following  differential  equations  will  furnish  all  the  partial  differential  coefficients 
required,  without  recourse  to  differentiating  the  equation  of  the  particular  problem  considered  : 

A,  a,  b,  c.    da  =  cos  C db  -\-  cos  B dc  -j-  sin  b  sin  C dA (28) 

B,  a,  b,  c.    db  =  cos  A  dc  -j-  cos  C  da  -(-  sin  c  sin  A  dB (29) 

f,  0,  b,  c.     dc  =  cos  B  da  +  cos  ^4  db  -\-  sin  «  sin  B  dC. (30) 

Or,  corresponding, 

t,  k,  d,  L.     dh  =  cos  q  dd  -\-  cos  ZdL  —  cos  d  sin  q  dt (31) 

Z,  h,  d,  L.     dd  =  cos  t  dL  -f-  cos  q  dh  —  cos  L  sin  /  dZ. (32) 

q,  k,  d,  L.    dL  =  cos  Z  dh  -f-  cos  t  dd  —  cos  /z  sin  2Td^ (33) 


1st.  Error  in  /.        ,   ,  -. 


LOCI  IN   TiME-ALTITUDE-LATITUDE-AZIMUTH. 

t,  h,  L,  to  find  Z ; 

A,  a,  c,  to  find  B.  Group  (4) 

B 


sin  B  cot  ^4  =  sin  c  cot  «  —  cos  c  cos  .Z?.  Eq.  (4) 

sin  Z  cot  ^  =  cos  L  tan  ^  —  sin  L  cos  Zl 


94  APPENDIX. 

From  (28),  o  =  cos  C  db -\-  sin  b  sin  CdA.     ...     0     ...     (34) 

From  (29),  db  =  sin  c  sin  A  dB.     ...    .     .    .    .     .•    .    ,     .     (35) 

dB  sin  b  sin  C  sin  £ 

Eliminating  £0,  -r^  =  ~ = — ^ 7*=  - —     — 7^ (36) 

dA.       sin  c  sin  yi  cos  o       sin  #  cos  C 

^?  =       cos^  (.7) 

dfr  COS  //  COS  <^*  S**' 

It  is  obvious  that  the  curve  of  elongations  is  that  of  absolute  max. 

Putting  d  (2d  member)  =  o,  eliminating  dh  and  dq,  and  simplifying,  we  obtain 

sin  Z  (cos  d  cos  q  sin  h  -j-  cos  L  cos  2T)  =  o (38) 

sin  Z  =  o,  the  meridian  one  branch. 

The  other  branch  by         sin  b  cos  C  cos  #  -j-  sin  c  cos  B  =  o (39) 

To  turn  into  B,  a,  c,  substitute  from  (17)  sin  b  cos  C, 

sin  a  cos  «  cos  c  —  cos2  #  sin  c  cos  .#  -j-  sin  c  cos  ^  =  o (40) 

Substitute  cos2  #  =  I  —  sin8  a. 

sin  c  sin2  #  cos  B  =  —  sin  #  cos  #  cos  c (41) 

cos  £  =  -r-  cot  <2  cot  c (42) 

cos  Z  =  —  tan  A  tan  L.     Equator (43) 

No  novel  curve,  and  (43)  proves  to  be  a  curve  of  constant  error;  for,  when  d  =  o,  by  substi- 

.    .  dZ          i 

tuting  (19)  in  (37)  we  have  -r-  =  — — 7-. 

dt       sin  L 

Meridian  for  algebraic  max.  and  min.  gives  numerical  min.  at  that  culmination  of  the  star 
having  the  less  altitude,  and  numerical  max.  for  greater  altitude.  But  both  these  are  numeri- 
cal min.  for  stars  whose  ±  d  >  Z,  as  compared  with  absolute  max.  for  q  —  90°,  270°.  The 
equator  divides  the  meridian  into  branches  of  algebraic  max.  and  min. 

7  D 

2d.  Error  in  L.      t,  /i,  j  ;  A,  a,  ^. 

By  (28)  and  (29),  db  =  cos  A  dc  +  sin  c  sin  A  dB; (44) 

o  =  cos  C  db  -\-  cos  B  dc .  (45) 

Eliminating  db, 

cos  C  cos  A  dc  -f-  cos  C  sin  c  sin  A  dB  -(-  cos  B  dc  =  o ;    „     .     .     .     .  (46) 

dB  _        cos  C  cos  A  -\-  cos  B 
dc  cos  C  sin  c  sin  A 

sin  C  sin  A  cos  b  ,     . 

and  by  (7),  =  -       —^-r : — z,    .     .     .     ........     (47) 

cos  C  sm  c  sin  ^ 

^2T      tan  q  sin  <a?  ,  ox 

or  ~TJ  =  -       — r — .     -    (48) 

*/£  cos  Z 


APPENDIX.  95 

By  inspection,  q  =  90,  curve  of  elongation,  absolute  max. ; 

q  =  o,  meridian,  absolute  min. ; 

Z  =  90,  prime  vertical,  algebraic  max.  and  min.,  giving  numerical  max. 
No  novel  curve. 

Z  B 

3d.  Error  in  h.     t,  L,  -7  ;  A,  c,  -. 

ft  Ct 

By  (28)  and  (29),       da  =  cos  Cdb  \ (49) 

db  =  cos  C da  -f-  sin  c  sin  AdB;    .     .     . (50) 

dB  _  sin"  C  _  sin  C  tan  C      tan  C* 

'  da  ~  sin  c  sin  A  cos  £  ~~  sin  c  sin  ^4  "~  sin  a  ' 

dZ           tan  ^  sin  q 


dh  cos  ^  cos  q  cos 


(52) 


By  inspection,  the  meridian  for  absolute  min.  and  curve  of  elongations  for  absolute  max. 
Algebraic  max.  and  min.  by 

/  sin  q      \ 

d\  --          —  j)=o  ............    (53) 

\      cos  q  cos  til 

COS*  q  cos  hdq-\-  sin"  q  cos  h  dq  -\-  cos  q  sin  q  sin  h  dh  =  o  ;      .     .     .     .     (54) 
cos  h  dq  -\-  cos  q  sin  q  sin  hdk  =  o. 

sin  ^  cos  Z. 


Substitute  d%  =  tan  Z  cos  /«  #<7,         and         sin  q  =  =• 

cos  d 

cos  d  cos  Z  -\-  cos  ^  sin8  Z  cos  L  sin  ^  =  o  ........     (55) 

As  an  example,  we  shall  give  the  work  instead  of  merely  indicating  steps. 

sin  b  cos  B  -f-  cos  C  sin2  B  sin  c  cos  «  =  o  ........     (56) 

Substitute  sin2  B=\  —  cos3  B  : 

sin  b  cos  ^+  cos  C  sin  £  cos  «  —  cos  C  sin  £  cos  a  cos2  B  =  o  .....     (57) 


cos  £  —  cos  a  cos 

Substitute  cos  C  =  -  :  --  -  —  7  ---  ,  from  (15), 

sm  a  sin  b 

and  clearing, 

sin*  b  sin  a  cos  .Z?  -|~  sin  c  cos  £  cos  a  —  sin  £  cos2  a  cos  #  | 

—  sin  c  cos  a  cos  c  cos2  .#  -f-  sin  c  cos2  #  cos  £  cos2  B  —  o.  f 


96  APPENDIX. 

Substitute  I  —  cos2  b  =  sin2  b: 

sin  a  cos  B  —  sin  a  cos2  b  cos  B-\-  sin  £  cos  c  cos  #  —  sin  c  cos*  #  cos  # 
—  sin  c  cos  #  cos  c  cos8  Z?  -f-  sin  c  cos4  #  cos  b  cos2 


z  cos  b  ) 

=  o.     f 


(59) 


Substitute  cos  b  =  cos  c  cos  #  -f-  sin  c  sin  #  cos  B, 

and          cos"  £  =  cos2  c  cos"  a  -\-  2  sin  £  sin  #  cos  £  cos  a  cos  .5  -j-  sin"  c  sin*  #  cos2 

and  we  have 

sin  a  cos  Z?  —  sin  a  cos2  £  cos2  «  cos  B  —  2  sin2  a  sin  c  cos  r  cos  a  cos2  Z? 

—  sin3  a  sin2  ^  cos3  B  -j-  sin  <:  cos  £  cos  #  —  sin  c  cos*  #  cos  £ 

—  sin2  c  sin  #  cos2  a  cos  Z?  —  sin  c  cos  £  cos  a  cos2  5 
-f-  sin  c  cos  £  cos8  #  cos2  B  =  o. 


(60) 


Collecting  coefficients  of  powers  of  cos  B, 


—  sn  a  sn  c 

-}-  sin2  c  sin  «  cos2  # 


cos8  B 

—  2  sin2  a  sin  £  cos  £  cos  a 

—  sin  £  cos  a  cos  £ 
-f-  sin  c  cos3  «  cos  c 


cos  Z? 


-f-  sin  a 


—  sn  #  cos  c  cos 

—  sin2  c  sin  #  cos2 


-f-  sin  £  cos  £  cos  a 
—  sin  £  cos3  a  cos2  c 


sin3  c  sin  #(cos2  #  —  sin2  a)  cos3 .5  —  3  sin  ^  cos  £  cos  a  sin2  #  cos2  B 

-f-  sin3  #  cos  Z?  -}~  sin  ^  cos 


c  cos  a  sin2  a  cos2  B  } 
DS  £  cos  a  sin2  #  =  o.  ) 


Dividing  by  sin  #, 


sin2  i:  (cos2  a  —  sin*  a)  cos3  B  —  3  sin  £  cos  c  cos  0  sin  a  cos2  .Z?  -j-  sin"  a  cos  .Z? 

-f-  sin  c  cos  c  cos 


sin  a  cos  B  \ 
a  sin  #  =  o ;  f 

os  Z  | 

=  0.    ) 


cos2  Z  (sin2  ^  —  cos*  h)  cos3  Z  —  3  cos  Z  sin  Z  sin  ^  cos  h  cos2  ^4~  cos2  ^  cos 

-J-  cos  Z  sin  Z  sin  /&  cos 

Let  B  represent  cos  Z,  whence  V~}~~-  B2  will  represent  sin  Z. 
Then,  from  article  (55)  preceding, 


(61) 

(62) 
(63) 


B2 


(i  -  rj 
(i  +  r2)2 


(I  + 


f  +  B  4/1  -  B2  X  ; 


X 


=  o.  . 


(64) 


G.C.D., 

...  B2(i  - 


—  36  4/1  —  B2  X  2ra  (i  —  r2)/ 
-f  4r> 


B  l/i  -  B2  (i  -  r2)  2r4  =  o. 


(65) 


By  -  6Bvy  +  Bvy 

-  6B  4/1  -  BV/  +  6B  4/i  -  B1  X  r* 


2B4/I  -Bfxr4—  2B  i/ 1  -  B2  X  r8  =  o. 


APPENDIX. 


Substituting 

and 

and  we  obtain, 


-f  / 


-  6B'>V  -  6B2/  +  B>V  +  2BV/  +  B2/  -  6B  i/i  - 
-  6B  Vr 


6B  I/  1  -  B>V  +  I2B  1/i^ 


+  6B  1/i  -  B2/  +  4*>  -f  8#y  +  4/  +  2B  Vi  -  BV 


+  46  l/i  -  BV/  +  2B  Vi  -  By  -  2B  4/i  -  BV-6B  i/i  -BV/ 


-  6B  i/r^^BV/  -  2B  i/7^=~By  =  o 
Collecting  coefficients  of  /,  /,  etc., 


97 


(65) 


+  6B  4/1  -  B3 

-  6B3 

-    6B  4/i  -  B3 

+       "D* 
J3 

-  6B  4/1  -  BV 

+  4*' 

-  2B  4/i  -  Ba 

-f  2BV 

4-  I2B  4/i  --  BV 

-  6BV 

4-  6B  4/i  —  BV 

(66) 

+  4 

+    2B  4/1  --  B3 

4-    BV 

4-  46  4/1  —  BV 

6B  4/i  -  BV 

4-  8*' 

-  6B  4/1  -  BV 

2B  ^i  -  BV, 


26  V i  -BV, 


absolute  terms, 
reducing, 

B1/  4-  48  4/i  -  B3/  -  6B3/  4-  2BV/  4-  4/  -  46  4/1  -  B'/ 
4-  6B  4/i  -  BV/  4-  B3/  -  6B3*3/  4-  BV/ 


(67) 


—  2B  4/i  —  BV/  4-  4*>  4-  2B  4/i  —  BV  —  2B  4/i  —  BV  =  o. 
Dividing  by  the  coefficient  of/,  and  arranging  in  order  of  terms  of  highest  degree, 


2      .     5       ,             4      3        ,4  < 

/i  —  Ba  e 

6  4/i  —  B2  „ 

,       24/1  --B3   6 

*>  +  */-«* 

B 

>*  +  ~*y 

/rirsv/ 

t~       g     x  y 

4---S 

B 

4   i   -t/3  —  o 

44/1  --  B3  4       24 

2  4/1  -  B3 

B 

B 

B 

i  ^  —  u> 

.     .     (68) 


which  is  the  equation  to  the  stereographic  projection  of  the  locus  of  algebraic  max.  and  min. 
for  error  in  h  in  time-alt.-lat.-azimuth. 


8. 


TIME-SIGHT. 


Given  h,  L,  ««*/  d,  ^  find  t. 

cos  #  =  cos  b  cos  £  4"  s'n  ^  sm  ^  cos  ^  • 
sin  ^t  =  sin  d  sin  Z.  4~  cos  d  cos  Z  cos  /; 


.     .     (69) 


98  APPENDIX. 

from  which  are  derived  the  well-known  formulas, 


/cos  s  sin  (s  —  ft)  .  ,  ,  .  T  ..  /' 

=  y       Cos  Z  sin/      '        m  which        *  ==l(£-f  / -f  *)?    •     •     •     (?o) 


/sin  j  sin  fa  —  z) 
=  V    sinco-Xsin/        m  wh'ch 


-*  /.ji****.^*.**     i  »*  *•'  /  •  1*1  1      /  7"         I          *.        l          7  \  /\ 

cos  ^  =  A  /  -         -,    .     /,        m  which        s  =  i  (co-Z  +/  +  A);      .     .     (71) 


cos  ^  sin      —  *  i  /  r    i    /    i 

ru^T)75rTr-^)'  *  =  *(£+*+/).   .  (72) 

Small  errors  in  the  data  will  have  the  same  effect  on  the  hour-angle  computed  by  each 
of  these  formulas.  So  far  as  inexactness  in  the  logarithmic  tables  is  concerned,  (72)  will  give 
the  result  nearest  to  precision. 

If  t  >  90°,  (71)  will  be  preferable  to  (70). 
If  t  <  90°,  (70)  will  be  preferable  to  (71). 

9.  ist  Case.     Error  in  t  owing  to  error  in  h. 

t  A 

Group  (5),  d,  L,  ^  ;  b,  c,  - . 

Differentiating  (69),  —  sin  a  da  =  —  sin  b  sin  c  sin  A  dA  ;   .     .     .     .     .     .     .     .     (73) 

dA  sin  a  sin  b  i 


da  '    sin  A  sin  b  sin  c       sin  B  sin  b  sin  c       sin  B  sin 
dt  i  i  cos 


—.;••••  (74) 

rr-     •    (75) 


d%  sin  ^  cos  Z  sin  q  cos  d?  sin  t  cos  </  cos  Z 

Inspection  of  (75)  shows  that  for  a  small  error  in  h  the  most  favorable  position  of  the 
body  is  either  on  the  prime  vertical  or  at  elongation,  whichever  is  attained;  hence  the  nearer 
in  bearing  to  the  prime  vertical  the  better.  At  a  given  place,  all  bodies  that  cross  the  prime 
vertical,  if  seized  exactly  on  it,  will  give  the  same  value  to  the  numerical  min.  whatever  the 
declination ;  and  the  less  the  latitude  the  better. 

If  the  body  does  not  cross  the  prime  vertical  the  best  position  is  when  at  elongation  ;  and 
the  less  the  declination  the  better.  For  a  given  body,  seized  exactly  at  q  =  90°,  or  270°,  all 
latitudes  permissible  (Z  <  d}  will  give  the  same  value  to  the  numerical  min. 

For  —  d  the  most  favorable  position  is,  theoretically,  in  the  horizon. 

If  observed  at  the  same  distance  from  the  meridian,  in  bearing,  on  both  sides,  the  con- 
stant error  in  h  (instrumental)  will  be  eliminated  ;  therefore  the  value  of  the  method  of  equal- 
altitudes  (with  a  correction  for  change  of  declination  when  there  is  any  change). 

The  most  unfavorable  position  is  on  the  meridian  giving  absolute  max. 


APPENDIX.  99 

10.  Inspection  gives  all  that  is  needed,  but  we  may  easily  deduce  the  algebraic  max.  and 
min.,  giving  the  numerical  min.  This  is  interesting,  as  giving  both  the  curve  of  elongations 
and  the  prime  vertical  ;  and  since  we  do  not  need  the  equation  for  curve-tracing,  let  it  be 
found  in  terms  of  t,  Z,  d. 

From  (75),  since  L  and  d  are  constant, 

d 


/      cosh\ 

\—   -r—  J=OJ    ............     (76) 

\       sin  t> 


sin  t  sin  h  dh  -)-  cos  h  cos  t  dt  =  o ; (77) 

dt  sin  /  sin  h  —  cos  h 

"  dh  ~        cos  t  cos  h  ~~  sin  /  cos  d  cos  Z  ^  ^      ' '  ^  ' 

/.  sin8  t  sin  ^  cos  Z  cos  d  —  cos  £  cos8  h  =  o (79) 

From  trig,  substitute,  i  —  sin2  h        for        cos2  k, 

and  sin  Z  sin  d-\-  cos  Z  cos  ^  cos  t        for        sin  & 

Performing  the  operations  and  collecting  terms, 

sin  Z  sin  d  cos  Z  cos  </  cos2  t  -\-  (cos2  Z  cos2  d-\-  sin2  Z  sin2  ^  —  i)  cos 


-{-  cos  Z  cos  d  sin  Z  sin  aT  =  o.  J 
For  coefficient  of  cos  t  substitute 

» 

—  sin2  Z  —  cos2  L        for        —  i, 
and  we  obtain  [cos2  Z  (cos2  d  —  i)  -(-  sin2  Z  (sin8  <aT  —  i)]  cos  t ; 

.*.  [—  cos2  Z  sin2  ^  —  sin2  Z  cos8  </]  cos  t. 
Substituting  this  in  (80),  and  dividing  by  the  coefficient  of  cos8  t, 

(  tan  d      tan  Z  ) 

cos8*-  -N    -F+. j}-cost=-i (81) 

(  tan  Z   '   tan  d  j 

Solving  (8 1 )  as  a  quadratic, 

/tan  d  ,    tan  Z\2       „  /tan2  ^  tan2  Z\ 

cos8 1  —          cos  /?  +  1 1 7  +  -    -J  =  1 1       ,  ,-  +  2  -f  - — j—il  —  i 

*  \tan  Z       tan  0/        *  \tan  Z.   '        '   tan   d' 

/tan8  d           .   tan2  Z' 

—  1  I 2 

tan  d  ,   tan  Z\  ..  /tan  </      tan  Z 


ioo  APPENDIX. 

Taking  upper  sign  of  second  member, 


cos  /  =  -^  —  j,  which  is  for  Z  =  90°  when  d  <  L  ........     (84) 

Taking  lower  sign, 

tan  L  _ 

cos*  =  -    —  -,,  f  or  q  =  90  °  when  d  >  L  ...........     (85) 

Lclll    w^ 

Though  in  the  problems  of  azimuth  both  of  these  have  been  found  to  be  branches  of 
numerical  min.  at  the  same  time,  yet  both  have  never  been  derived  algebraically  from  the 
given  expression  for  error. 

II.  2d  Case.  —  Error  in  t  owing  to  error  in  L. 

,     ,  t  A 

n,  a,  -j^',  a,  b,  -  . 

From  (69)  we  obtain 

o  =  —  cos  b  sin  c  dc  -J-  sin  b.  cos  c  cos  A  dc  —  sin  b  sin  c  sin  A  dA.    .     .     .     (86) 

dA  cot  b  sin  c  —  cos  c  cos  A 

Dividing  by  sin  b  ---  ~  ~'    ........ 


sin  A  cot  B  cot  B 

And  from  (2),  =  --  :  --  =—  ^-  =  --  :  --  ;    .........     (88) 

sin  c  sin  A  sin  c 

dt  i  dL 

.'.  -77  =  r  -  ^  --  T  dt^  =  7  -  ^  --  T  .......      (oQ) 

dL       tan  Z  cos  L  tan  Z  cos  L 

Inspection  shows  that  the  prime  vertical  and  the  curve  of  elongations  form  the  locus  of 
minimum  errors  in  the  computed  time  ;  the  former  for  absolute  min.  when  ±  d  <  L,  and 
the  latter  for  algebraic  max.  and  min.,  giving  numerical  min.,  when  ±  d  >  L.  It  will  not 
be  necessary  to  derive  the  equation  to  the  latter.  The  lower  the  latitude  the  better. 

12.  ^d  Case.  —  Error  in  /  owing  to  error  in  d. 

t  A 

'     '  ~d  '  a'  c'  ~b  ' 

Interchanging  b  and  c,  d  and  L  in  the  preceding,  we  derive 


j* 

dtd  =  -  -        —  ............     (90) 

tan  q  cos  d 


APPENDIX.  1 01 

Inspection  shows  that  the  locus  of  numerical  min.  consists  of  the  same  branches  as  for  error 
in  L ;  but  here  the  curve  of  elongation  gives  absolute  min.  when  ±  d  >  L ;  and  the  prime- 
vertical  gives  numerical  min.  from  algebraic  max.  and  min.,  f or  ±  d  <  Z,. 


Total  error,   dt  = : — ~ 7 dh  4-  -  —- = ~dL  -f-  -  — ;</<:/.    ....     (91) 

sin  Z^  cos  L        '   tan  Z  cos  Z  tan     cos  d 


13.  An  erroneous  assertion  by  Chauvenet  should  be  corrected. 
Chauvenet's  Astronomy,  vol.  i.,  page  212,  says: 


dd 

"dt=-    —,- , (92) 

cos  d  tan  q 

which  shows  that  the  error 

in  the  declination  of  a  given  star  produces  the  least  effect  when  the  star  is  on  the 


[i] 

(      prime  vertical ; 

[2]     and  of  different  stars  the  most  eligible  is  that  which  is  nearest  the  equator." 

(a)  Now,  as  far  as  error  in  declination  is  concerned,  if  we  may  select  the  star  to  be  given, 
we  should  take  one  having  d  >  L  (which  will  not  come  on  the  prime  vertical)  and  observe 
at  greatest  elongation,  q  =  90°,  when  dtd  =  o.  If  observed  at  this  instant,  it  matters  not  what 
the  declination  may  be,  between  the  limits  d  =  90°  and  d  >  L ;  but,  for  failure  to  seize  the 
star  exactly  at  q  =  90°,  the  less  the  declination  the  better. 

(#)  Supposing  the  star  given  us  does  cross  the  prime  vertical,  then  [i]  is  true  but  [2]  is 
false. 

(c]  Since  errors  in  d  are  less  likely  to  occur  than  errors  in  L  and  h,  and  since  these  two 
errors  will  give  the  least  dt  when  Z=  90°,  we  would  better  select  a  star  that  crosses  the 
prime  vertical,  when  dth  =  cosec  Z  sec  L  dh  =  sec  L  dh,  and  dtL  =  sec  L  cot  Z  dL  =  o. 

(d}  Therefore,  for  error  in  h  and  error  in  L  it  matters  not  what  the  declination  is,  pro- 
vided the  star  is  observed  on  the  prime  vertical. 

(e)  But  for  error  in  d,  the  most  eligible  star  of  those  that  cross  the  prime  vertical  is  that 
having  greatest  declination, — not  as  asserted  in  [2]  preceding. 

(/)  Therefore,  for  errors  in  all  the  data,  theoretically,  the  most  eligible  star  is  the  one 
crossing  the  prime  vertical,  that  has  the  greatest  declination  ;  that  is,  d  =  L,  making  Z  =  90° 
and  q  =  90°  when  the  star  is  on  the  meridian.  Practically  we  should  avoid  stars  on  the  me- 
ridian and  take  one  whose  d<L. 


14.  Summary  and  Proof  of  Fallacy. 

ist.  The  most  favorable  star  when  the  error  in  declination  alone  is  concerned  is  any  one 
whose  d  >  Z.,  provided  it  is  observed  at  q  =  90°,  then  dtd  =  o.  This  excludes  stars  that 
cross  the  prime  vertical. 

2d.  But  Chauvenet  makes  his  given  star  come  on  the  prime  vertical.     Therefore  d  <  L, 


102  APPENDIX. 

and  q  cannot  =  90°;  the  error  dtd  cannot  reduce  to  o,  but  it  will  be  least  for  q  nearest  to  90°, 
which  will  occur  when  Z  =  90° ;  that  is,  when  the  star  is  on  the  prime  vertical. 

3d.  Of  several  given  stars  observed  on  the  prime  vertical  the  most  favorable  one  will  be 
that  whose  cos  d  tan  q,  when  on  the  prime  vertical,  is  greatest.  And  this  will  be  for  d  =  L. 
For  notwithstanding  the  greater  d  is,  the  less  the  cos  d,  yet  the  greater  the  d  the  nearer  will 
q  be  to  90°,  and  tan  q  will  be  the  greater;  and  tan  q  will  preponderate,  making  cos  d'by  its 
side  insignificant. 

Singularly  enough,  in  (92)  Chauvenet  gives  cos  d  overwhelming  preponderance  over  tan  q, 

provided  the  star  is  on  the  prime  vertical ;  whereas  in  dZ  = 7  dh,   of  like   form,  he 

tan  q  cos  h 

permits  cos  h  to  have  no  influence,  and  assigns  the  star  to  the  prime  vertical  on  the  strength 
of  tan  q  alone. 


Proof  of  Fallacy. 

Changing  the  form  of  Chauvenet's  expression   into  one  equivalent,  we  have,  since  sin  Z 
cos  L  =  sin  q  cos  d, 

dt  = —. — ^ Y  dd. .     .     .     (93) 

sm  Zcos  L 

Therefore,  for  any  given  star  the  nearer  q  and  ^both  are  to  90°  the  more  favorable;  and  as 
one  nears  90°  so  does  the  other. 

For  any  one  of  several  given  stars  restricted  to  d  <  L  and  to  observations  on  the  prime 
vertical,  (Z  =  90°), 


cos  L 


L  being  fixed,  to  select  the  best  one  of  these  stars  we  have  to  consider  q  alone,  and  take 
that  one  giving  q  nearest  to  90°  which  will  make  cos  q  the  least. 

sin  Z  cos  L 

Now,  since  sin  q  =  -         —j — , 

*  cos  d 


("*("\C     j 

when  Z—  90°,  sin  q  — -> (95) 

cos  d 

Cos  L  being  constant,  q  will  be  nearest  to  90°  when  the  declination  is  greatest ;  or 
directly  from  (95)  q  =  90°  when  d '=  L,  and  the  star  that  is  farthest  from  the  equator,  instead 
of  nearest  (as  asserted  by  Chauvenet),  is  the  most  eligible. 


APPENDIX. 


103 


15- 


LATITUDE  PROBLEM. 


Latitude  by  an  altitude  observed  at  any  time. 
Given  t,  h,  and  d,  to  find  L. 
Given  A,  a,  and  b,  to  find  c. 


cos  a  =  cos  b  cos  c  -f-  sin  b  sm  c  cos  A  ; 
sin  h  =  sin  ^/  sin  Z.  -|-cos  ^  cos  L  cos  /  ; 


(96) 


from  which  are  derived  the  well-known  formulas, 


tan  0  =.  tan  d  sec 

sin  0  sin 
cos  0   ==  — 

sm  # 

L  =  0  T  0'. 


(97) 


Attending  to  the  algebraic  signs  of  0  and  0',  there  will  still  be  two  values  of  L.     The 
proper  value  will  be  determined  by  the  known  approximate  value. 
16.   ist  Case.     Error  in  L  owing  to  error  in  h. 


From  (96),  A  and  b  being  constant, 


sin  a  da  =  —  cos  b  sin  cdc-\-  sin  b  cos  c  cos  A  dc ; 


(98) 


da  cos  b  sin  c  —  sin.#  cos  c  cos  A 

dc  sin  a 


(99) 


and  from  (16), 


sin  a  cos  .Z? 
sin  « 


=  cos 


(100) 


~j~ 
da 


cos 


dh 
cos  ^' 


By  inspection,  the  most  favorable  position  of  the  body  is  on  the  meridian,  and  the  least 
favorable  on  the  prime  vertical,  giving  absolute  max.  If  the  star  does  not  come  on  the  prime 
vertical,  the  least  favorable  position  will  be  on  the  curve  of  elongations,  q  =  90°,  270°.  The 
latter  condition  is  obvious,  but  it  may  be  found  algebraically,  there  being  algebraic  max.  and 
in  in. 


104  APPENDIX. 

17.  2d  Case.     Error  in  L  owing  to  error  in  /. 
This  is  the  reciprocal  of  (89), 


/.  —  =  tan  Z  cos  L,  ...........     (IO2) 


and  the  max.  and  min.  are  interchanged,  as  compared  with  art.  n. 
18.  3^  Case.     Error  in  L  owing  to  error  in  d. 


From  (96), 

o  —  —  cos  c  sin  b  db—  cos  b  sin  c  dc  -j-  cos  b  sin  c  cos  A  db  -f-  sin  b  cos  c  cos  A  dc  ;  (103) 
dc  sin  b  cos  c  —  cos  b  sin  c  cos  y4 


sin  c  cos  £  —  cos  c  sin     cos 


(104) 


and  by  (16)  and  (19), 

sin  a  cos  {7  cos  C 


(105) 

sin  a  cos  B  cos  B 


cos 


(106) 


The  meridian  is  found  to  be  the  locus  of  algebraic  max.  and  min.,  and  this  gives  numeri- 
cal max.  for  ±  d  >  L  and  numerical  min.  for  ±_  d  <  L,  since  for  q  =  90°  there  is  absolute 
min.  and  for  Z  =  90°  an  absolute  max. 

In  the  curves  discussed  in  this  treatise  heretofore,  q  =  90°  and  Z=  90°  have  been  com- 
panions very  often  in  giving,  alike,  numerical  max.  or  min.  But  here  they  part  company, 
one  giving  absolute  min.,  the  other  giving  absolute  max. 

19.  THE  COMPUTATION  OF  THE  ALTITUDE. 

Given  t,  L,  d,  to  find  h. 

(a)  For  error  in  h  owing  to  error  in  t  we  have  the  reciprocal  to  (75)  and  an  interchange 
of  max.  and  min. 

(b)  For  error  in  h  owing  to  error  in  L  we  have  the  reciprocal  to  (101)  and  max.  and  min. 
interchanged. 

(c)  For  error  in  h  owing  to  error  in  d, 

-  a 

t,  L,,      •  A,  C,  Y. 

a  n 


APPENDIX.  I05 

Differentiating  (69), 

\  t 

—  sin  a  da  =  —  cos  c  sin  b  db  -}-  sin  £  cos  A  cos  #  d# ; (IO7) 

/ 

dfo       -j-  cosV  sin  b  —  sin  c  cos  #  cos  A 


db  -\-  sin  # 


(108) 


, .     /    »                                    sin  #  cos  C 
and  by  (19),  = — -5—-  =  cos  C. (109) 


Therefore  absolute  min.  when  q  =  j    9°0  [  and  numerical  max,  when  q  =  |    8°0  I  from  al- 
gebraic max.  and  min.,  the  maximum  being  equal  to  unity. 

Numerical  min.  >  o  and  <  i  for  Z  =  90°  from  algebraic  max.  and  min. 

Loci.     The  curve  of  elongations  and  the  prime  vertical  for  min.     The  meridian  for  max. 


EXPLANATION  OF  THE  PLATES. 


The  numbers  shown  on  the  plates  correspond  to  the  loci  numbered  in  the  text. 

Each  locus  is  shown  separately  for  the  latitudes  of  30°,  45°,  and  60°,  marked  respectively 
A,  B,  and  C. 

Each  plate  shows  the  stereographic  projection  of  the  sphere  on  the  plane  of  the  horizon  ; 
the  prime-vertical  being  the  axis  of  x,  and  the  meridian  that  of  y. 

On  each  plate  are  drawn  diurnal  circles,  i.e.,  parallels  of  declination,  marked  at  their  inter- 
sections with  the  meridian,  the  axis  of  y,  as  follows  : 

d'  for  a  star  having  -f-  d  >  L ; 

d"  for  a  star  having  -f-  d  <  L ; 
d'"  for  a  star  having  —  d  <  L  • 
dlv  for  a  star  having  —  d  >  L. 

Diurnal  circles  are  shown  also  for  d  =  o,  the  equator ;  and  for  the  parallel  —  d  =  L,  as  a 
right  line  passing  through  the  nadir  at  infinity,  and  parallel  to  the  prime-vertical,  the  axis  of  x. 

The  diurnal  circle  -f-  d  =  L  is  not  drawn,  but  it  may  readily  be  imagined  as  tangent  to 
X  at  the  zenith,  Z. 

The  astronomical  triangles  are  not  constructed,  since  their  lines  would  encumber  and  ob- 
scure the  plates  for  their  more  important  use,  that  of  showing  the  curves  of  max.  and  min. 
errors.  The  triangles  may  easily  be  completed,  mentally,  by  imagining  the  vertical  circles, 
through  Z,  and  the  hour-circles,  through  P,  as  intersecting  on  the  parallels  of  declination  at 
the  points  discussed  as  giving  max.  and  min.  errors — thus  with  co-Z,,  already  drawn,  forming 
the  triangle. 

Lines  above  the  horizon  are  drawn  full ;  those  below,  dotted. 

The  loci  are  discussed  with  regard  only  to  the  theoretically  most  favorable  and  least  favor- 
able positions  of  the  body  (see  Introduction,  art.  3)  ;  and  equal  respect  will  be  paid  to  points 
below  the  horizon  as  is  paid  to  points  that  would  be  visible  to  the  observer  on  the  earth. 

Again,  indetermination  may  occur  at  the  point  that  otherwise  would  be  theoretically  the 
best,  and  the  close  vicinity  to  this  point  will  be  a  favorable  region. 

Therefore  the  point  that  theory  declares  to  be  the  most  favorably  situated  may  be  re- 
ferred to  one  that  \*  practicably  the  best  from  the  theoretical  point  of  view.  The  terms  prac- 
ticably and  practically  should  not  be  confounded,  as  the  best  practical  conditions  may  be  left 
for  the  observer  to  select,  after  presenting  those  that  in  theory  are  the  best. 

Since  true  maxima  and  minima  are  such  with  respect  to  the  values  of  the  function  that 
lie  immediately  on  each  side  of  them,  neither  the  several  maxima  being  equal  to  one  another, 
nor  the  several  minima  equal  to  one  another ;  and  since,  also,  of  alternate  true  maxima  and 


EXPLANATION  OF   THE  PLATES.  107 

minima,  a  particular  maximum  need  not  be  so  great  as  a  minimum  that  is  not  alternately 
near;  and  since,  moreover,  disregarding  signs,  out  of  several  numerical  maxima  of  the  func- 
tion, one  may  be  the  greatest,  and  one  the  least,  in  numerical  value  (and  the  same  respecting 
numerical  minima), — therefore  we  shall  avoid,  in  defining  the  points  in  the  first  instance,  un- 
less there  can  be  no  mistaking  them,  the  use  of  the  terms  most  favorable  and  least  favorable ; 
and  shall  say  favorable  and  unfavorable,  these  terms  corresponding  respectively  to  numerical 
min.  and  numerical  max.,  whichever  algebraic  sign  the  function  may  have.  The  terms  most 
and  least  favorable  may  afterwards  be  used  with  discrimination. 

In  no  case  of  these  loci  has  it  been  found  necessary  to  have  recourse  to  the  second  differ- 
ential of  the  expression  for  the  error,  in  order  to  discriminate  the  maxima  and  minima.  In 
No.  I  the  method  of  discriminating  is  given,  for  illustration  ;  but  it  is  deemed  unnecessary  to 
repeat  this  proceeding  in  detail  in  all  cases,  since  what  is  obvious  to  the  writer  will  be  obvious 
to  the  reader. 

A  positive  error  in  the  datum  is  assumed  for  the  discussion  of  maxima  and  minima  values 
in  the  error  of  the  computed  azimuth.  Evidently,  to  make  the  expressions  serve  in  practice  as 
corrections,  the  proper  sign  of  the  error  found  to  have  existed  in  a  given  part  must  be  used  ; 
and,  also,  the  sign  of  the  expression  for  the  error  must,  itself,  be  changed. 

It  will  be  necessary  to  keep  in  mind  the  method  used  in  this  treatise  in  reckoning  the 
angles  /,  Z,  and  q,  and  the  explanation  of  the  terms  maximum  and  minimum,  as  here  used. 
The  reckoning  of  Z  differs  from  that  of  astronomers,  generally,  who  reckon  from  the  south 
point  of  the  horizon.  But  the  writer  considers  the  method  used  in  this  treatise  the  rational 
one  for  the  purpose  in  view ;  and  there  seemed  no  way  of  escape  from  making  terms  with  the 
maxima  and  minima.  (See  pp.  19-21.) 

Though  P  in  the  diagrams  may  be  either  the  north  or  south  pole,  whichever  is  the 
elevated  pole,  so  long  as  d  of  the  same  name  as  the  latitude  is  regarded  as  positive,  and  if 
of  the  contrary  name  as  negative,  yet,  for  convenience  and  uniformity,  the  plates  are  for 
north  latitude.  Therefore  with  respect  to  the  plates  we  refer  to  points  north  and  south,  as 
well  as  east  and  west. 

In  referring  to  circumpolar  stars  we  may  have  those  with  d  <  L,  possible  only  when 
L  >  45°;  one  with  d  =  X,  =  45°  as  the  single  case ;  those  whose  d  >  L,  which  may  be  called 
close  circumpolar  stars,  for  they  will  possess  ihe  general  characteristics  of  stars  very  near  P, 
/  being  small,  and  they  cannot  have  a  d  <  45°. 

There  will  be  stars  to  consider  whose  d  >  Z,  yet  not  circumpolar  stars,  possible  only 
when  L  <  45°;  and  the  stars  whose  d  <  L  cannot  be  circumpolar  ones  when  L  <  45°. 

In  referring  to  the  plates,  only  a  brief  of  the  matter  from  the  text  will  be  given,  and 
abbreviations  will  be  largely  used. 


Locus  No.  i. — ALTITUDE-AZIMUTH  ERROR  IN  h. 

[Summary  of  pp.  25,26,31-36,55-63-]        §  =  ^ — ^ (55) 

Branches:     I.  Absolute  max.,  the  merid.  NFS. 

2.  Absolute  min.,  curve  of  elongations  Pa' Za"  and  e'P'e". 

3.  Num.  min.,  from  alg.  max.  and  min.,  curve  c'Eb'Zb"  We". 

It  is  obvious  that  the  merid.  and  q  =  90°  curves  give  no  alg.  max.  or  min.;  for  the  sign 
of  (55)  depends  on  tan  q,  since  cos  h  is  always  positive.  (55)  therefore  changes  sign  when  q 
passes  through  the  values  o°,  90°,  180°,  270°. 

At  b",  W,  and  c",  q  lies  between  o°  and  90°;  therefore  tan  q  is  (-}-),  and  alg.  max.  occur 
with  respect  to  alg.  min.  at  b ',  £,  and  c1 ',  where  tan  q  is  (— ),  since  q  >  270°  and  <  360°. 
But  on  a  given  par.  of  dec.  these  alg.  max.  and  min.  are  equal  numerically,  and  both  are 
num.  min.  as  compared  with  abs.  max.  on  merid. 

Following  the  star  in  its  diurnal  course,  beginning  at  lower  cul.,  we  find  : 

For  -j-  d  >  L,  at  d\  abs.  max.  oo  ;  a ',  abs.  min.  o  favorable ;  d',  abs.  max.  oo  ;  a",  abs. 
min.  Q  favorable. 

For  -\- d  ==•  L,  at  low.  cul.  abs.  max.;  at  Z  indeterminate,  but  akin  to  a  min.  The  slight- 
est departure  from  the  zenith  gives  a  determinate  case,  and  a  very  small  error,  which  increases 
continuously  during  the  progress  of  the  star  towards  low.  cul.,  where  oo  occurs. 

For  -\- d  <  Z.,  at  low.  cul.  d",  abs.  max.  oo  ;  b',  alg.  min.,  num.  min.  (  — ),  favorable;  d", 
abs.  max.  oo  ;  b",  alg.  max.,  num.  min.  (-[-).  favorable. 

For  d=  o,  on  merid.  abs.  max.  oo ;  at  E  and  W,  num.  min.,  (— )  and  (-J-)  respectively, 
favorable. 

For  —  d  <.  L,  num.  min.  at  c'  (— )  and  c"  (-J-),  favorable,  but  practicably  in  the  horizon. 

For  —  d=  L.  In  the  projection,  this  is  the  asymptote  to  the  curve  of  alg.  max.  and 
min.,  meeting  it  as  a  tangent  at  infinity,  the  nadir.  Practicably,  the  horizon  gives  its  favor- 
able points. 

For  —  </>  L,  abs.  min.  o  at  e'  and  e" ,  favorable  ;  practicably  in  horizon,  if  star  rises. 

The  unlettered  line  parallel  to  —  d  =  L  is  the  asymptote  to  the  lower  curve  of  elonga- 
tions. In  the  projection  it  crosses  the  merid.  at  the  distance  2  tan  L  from  the  origin  ;  and, 
from  the  origin,  —  d  =  L  lies  at  the  distance  tan  L.  The  lower  asymptote  is,  on  the  sphere, 
a  small  circle  inclined  to  the  equator,  and  is  tangent  at  the  nadir  to  the  p.  v.  and  to  the  curve 
of  elong. 

For  all  bodies  whose  —  d>  L,  and  d=  o,  the  most  favorable  practicable  positions  are  in 
the  horizon.  The  constant  (instrumental  and  personal)  error  in  h  will  be  eliminated  in  the 
mean  of  the  results  of  observation  of  a  star  east  and  west  of  the  meridian,  if  taken  on  the 
same  parallel  of  altitude. 

The  algebraic  max.  and  min.  curve,  coinciding  with  the  p.  v.  in  L  =  o°,  swells,  with  in- 
creasing latitudes,  until  the  max.  ordinate  occurs  (or  widest  departure  from  the  p.  v.),  in  about 
latitude  54°  31';  it  then  returns  to  its  first  form  by  the  time  that  latitude  90°  is  attained. 

(Compare  with  No.  4.) 


Altitude -Azimuth,   error  ink 


Altitude -Azimuth,    error  in  L 


Altitude  Azimuth,   error  in  // 


s 


If... 


Locus  No.  2.  —  ALTITUDE-AZIMUTH  ERROR  IN  L, 

[Summary  of  pp.  26,36,37,63,64.]         d=  —J—.  .     .     .  (56) 


Branches:     i.  Absolute  max.,  the  merid.  NPS. 

2.  Absolute  min.,  the  six-hour  circle  Pa'b'E  .  .  .  .  P'  .  .  .  .  W  ....  P. 
Neither  branch  gives  algebraic  max.  and  min.,  because  tan  t  changes  sign  when  the  given 
body  transits  the  six-hour  circle  and  the  meridian. 

The  least  favorable  positions  for  any  star  are  those  on  the  meridian,  the  error  giving  <x>  . 
The  most  favorable  positions  are  on  the  six-hour  circle,  giving  o  for  error,  as  follows  : 


+  d> 

L 
L 
o 
L 
L 
L 

at 
at 
at 
at 
at 
at 

a'     and 
V     and 
E    and 
c'     and 
cutting 
e'     and 

But  for  all  rising  and  setting  bodies  having  negative  declinations  the  best  practicable 
points  are  in  the  horizon. 

In  determining  the  direction  of  the  meridian  on  shore,  the  error  in  latitude  will  be  elimi- 
nated in  the  mean  of  the  results  of  observation  of  the  given  body  east  and  west  of  the 
meridian  on  the  same  parallel  of  altitude. 


Alcilude  -Azimuth,    error  in  L 


Altitude  -Azimuth,    error  in  L. 


M2 


\liiludi'  -Azimuth,   error  in  L,. 


-7                                                                                1 

'                                                                                 1 

'                                                               1 
'                                                                          1 

f-d  =  LJ             { 

\ 
\ 
•  \ 
i 
i  • 
i 
i 
.-  i 

Locus  No.  3. — ALTITUDE-AZIMUTH  ERROR  IN  d. 

[Summary  of  pp.  26,  37,  64.]     —  = : — (57) 

1     dd  sin  /  cos  L 

Branches:     I.  Absolute  max.,  the  meridian  NFS. 

2.  Num.  min.,  from  alg.  max.  and  min.,  the  six-hour  circle  Pa'b'E  .  .  .  P',  etc. 

These  branches,  with  their-  kinds  of  max.  and  min.,  are  obvious  ;  for  sin  t  changes  sign 
on  passing  through  o°  and  180°;  but  does  not  change  sign  when  passing  through  90° 
and  270°. 

From  the  sign  of  (57)  we  see  that  the  num.  minima,  having  the  same  numerical  value 
east  and  west  of  the  meridian,  are  alg.  min.  on  the  west  side  and  alg.  max.  when  east. 
The  least  favorable  position  for  any  given  body  is  when  on  the  merid.,  error  =  oo, 
The  most  favorable,  the  error  having  a  finite  numerical  value  =  sec  Z,,  is  on  the  six-hour 
circle,  as  follows : 

-f-  d  >  L,  at  a'  or  a"  ; 
-f-  d  =  Z,  at  points  on  the  6h  circle  ; 
+  d  <  L,  at  V  or  b"  ; 
d  =  o,  at  E  or  W; 

—  d  <  L,  at  c'  or  c"  ; 

—  d  =  Z,  at  points  on  the  6h  circle ; 

—  d  >  Z,  at  e'  or  e". 

The  points  for  max.  and  min.  are  the  same  as  in  No.  2,  but  the  num.  min.  in  No.  2  is 
absolute,  o,  for  all  bodies  ;  while  in  No.  3  it  is  equal  to  sec  L  for  each  of  all  bodies. 

For  all  rising-and-setting  bodies  having  negative  declinations,  the  best  practicable  points 
are  in  the  horizon. 

If  the  given  body  is  observed  east  and  west  of  the  merid.  on  the  same  par.  of  altitude, 
the  error  will  be  eliminated  in  the  mean  of  the  results  of  computing  the  azimuth  ;  and  since 
this  condition  is  the  same  regarding  error  in  h  and  error  in  L,  we  should  observe  as  follows : 

If  the  error  in  L  is  likely  to  have  a  more  pernicious  effect  than  the  error  in  h,  owing  to 
uncertainty  of  the  geographical  position  while  our  instrument  and  the  conditions  for  observa- 
tion of  the  altitude  are  good, — we  should  observe  the  star  east  and  west,  on  the  six-hour 
circle :  then,  failing  to  seize  the  observation  exactly  at  the  points  aimed  for,  the  failure  to 
eliminate  the  error  whoMy  wiirhot  have  much  effect. 

But  if  the  conditions  are  reversed,  and  error  in  h  is  more  to  be  apprehended,  try  to  seize 
the  body  east  and  west  when  on  the  curve  of  min.  error  in  No.  I. 

In  moderately  high  latitudes  a  close  circumpolar  star  will  give  q  —  90°  and  /  —  90°  very 
near  each  other,  so  that  whichever  is  chosen  it  will  be  favorable  for  eliminating  the  errors  in 
both  L  and  h,  and  also  d.  Besides,  the>akrtude~  will  be  favorable  for  the  practical  problem 
aside  from  the  theoretical. 


Altitude -A.zttnu.ih.   error  in  A 


Al/i/in/f   A'/J//iu//i,    ('/•/•<;/•  i//  d 


M.3 


B 


'•Altitude -Azimuth,   error  in.  rf 


Locus  No.  4. — TIME-AZIMUTH  ERROR  IN  /. 

,     dZ            cos  a  cos  d 
[Summary  of  pp.  26,  27,  37-41,  64-70.]     —  = — -— — (58) 

Branches:  i.  Absolute  min.,  the  curves  of  elongation  Pa'  Za"  and  e'P'e" . 

2.  Alg.  max.  and  min.,  giving  num.  max.  and  min.,  the  merid.  NFS  and  the  curve  c'Eb'Zb"  We"- 
The  latter  gives  always  numerical  min.,  and  pertaining,  as  it  does,  to  ±  d  <  L,  all  those  stars  will  cross  it 
unless  the  latitude  is  very  high.  So  long  as  all  stars  having  ±  d  <  L  cross  this  curve  c'E ....  c",  the  merid. 
will  give  num.  max.  at  each  culmin.  for  all  stars.  Because,  for  ±  d>  L  the  values  on  the  merid.  will  be 
numerically  greater  than  the  valge  o  on  the  curves  of  elong.  ;  and  for  ±  d  <  L  they  will  be  greater  on  the 
meridian  than  on  the  other  curve.  But  in  latitudes  >  70°  32'  (about)  some  ±  d's  <L  will  not  touch  the 
curve  c'E .  .  .  .  c" ;  therefore  we  must  look  to  the  other  branch  of  alg.  max.  and  min.  for  the  num.  min. 
This  will  occur  at  that  culmination  of  the  star  that  has  the  less  numerical  value  of  h,  whether  -|-  or  — .  To 
bring  this  about,  the  latitude  must  be  so  great  that  we  may  say  the  meridian  gives  numerical  max.  generally. 
In  (58),  since  cos  d  and  cos  h  remain  positive,  the  sign  of  the  expression  depends  on  the  sign  of  cosy,  depen- 
dent in  turn  on  the  quadrant  in  which  q  is  situated. 

Taking  -\-d>  L,  alg.  max.  and  min.  both  occur  on  the  meridian.     At  upper  transit,  q  —  180°,  cos^  does 

not  change  sign  when  passing  through  this  value,  but  is  (— )  on  each  side ;  therefore  the  sign  of  ~   is  (-f ) 

Ctrl 

and  we  have  an  alg.  max.  At  lower  transit  q,  having  passed  through  the  value  90°  (causing  cos  q  to  change 
sign,  and  so  giving  no  alg.  max.  or  min.,  though  the  value  o  is  found)  is  equal  to  oc ;  and  in  transit  cos  q 

7  r? 

does  not  change  sign,  but  is  (-f)  on  each  side,  making  the  sign  of  —7-  (— ),  whence  an  alg.  min.     To  attain 

upper  culmin.  again,  q  passes  through  the  value  270°,  cos^  changes  sign,  and,  though  we  get  o  for  the  error, 
no  alg.  max.  or  min.  occurs  until  the  max  on  the  merid.  recurs. 

Therefore  for  Js-d>  L,  the  num.  max.  and  min.,  if  governed  solely  by  algebraic  max.  and  min.,  would 
correspond  to  the  latter,  not  only  numerically  but  in  the  algebraic  signs ;  for  the  (+)  error  is  the  greater 
numerically  since  cos  h  at  the  upper  transit  is  less  than  at  the  lower.  But  at  a'  and  a"  absolute  min.,  o, 
intervene  and  the  alg.  max.  and  min. •  on  the  merid.  both  become  num.  max. 

For  —  d  >  L  similar  conditions  prevail,  but  not  exactly  like  conditions,  for  the  alg.  max.  and  min.  are 
reversed,  both  in  numerical  value  and  sign  ;  for  cos  h  is  then  the  greater  at  upper  transit,  and  q  there  passes 
through  o°,  and  at  lower  transit  through  180°. 

Taking  ±  d  <  L,  q  on  one  side  of  the  meridian  remains  always  between  o°  and  90°.  and  on  the  other  be- 

J  "7 

tween  360°  and  270°.     Therefore  the  sign  of  cosy  is  always  (-}-)  and  the  sign  of  — y  always  (— ).  Therefore  alg. 

max.  and  min.  occur  alternately  ;  the  minima  on  the  meridian  (unequal  numerically)  and  the  maxima  on  the 
other  curve  (equal  numerically).  But  our  numerical  maxima  and  minima  (unfavorable  and  favorable  points^ 
interchange  with  the  algebraic  minima  and  maxima. 

To  follow  the  star  in  its  diurnal  course,  beginning  at  the  lower  culmination  : 

For  -j-  d  >  L.  At  d',  alg.  min.,  num.  max.  (— ),  unfavorable ;  at  a',  abs.  min.  =  o,  favorable;  d ',  alg.  max., 
num.  max.  (+),  itnfavorable ;  a'1,  abs.  min.  =  o,  favorable. — For  -\-  d  <  L.  At  d'1,  alg.  min.,  num.  max.  (— ), 
unfavorable;  b' ,  alg.  max.,  num.  min.  (— },  favorable;  d" ,  alg.  min.,  num.  max.  (— ),  unfavorable ;  b" ,  alg.  max., 
num.  min.  (— ),  favorable.  —  For  -\-d-L.  Practicably,  favorable  very  near  Z,  the  zenith,  and  unfavorable  at 
low.  culm.  At  Z  the  value  is  indeterminate. — For  d  =  o.  At  E  and  W  favorable,  on  the  merid.  unfavor- 
able.— For  —  d  <  L.  At  d'" ,  low.  culm.,  alg.  min.,  num.  max.  (— ),  unfavorable;  at  c',  alg.  max.,  num.  min. 
(— ), favorable ;  at  d"' ,  alg.  min.,  num.  max.  (— ),  unfavorable;  at  c",  alg.  max.,  num.  min.  (— ),  favorable. — For 
—  d  >  L.  At  low.  culm.,  alg.  max.,  num.  max.  (-}-),  unfavorable;  at  e',  abs.  min.  =  o,  favorable ;  at  d IT, 
above  horizon,  alg.  min.,  num.  max.,  unfavorable  ;  at  e" ,  abs.  min.  =o,  favorable. 

For  d=  o  and  for  negative  d's  the  horizon  furnishes  the  best  practicable  points  if  the  body  rises  and  sets. 

The  error  in  Z  due  to  error  in  t  will  not  be  eliminated  by  observing  the  star  east  and  west,  symmetrically 
situated  with  respect  to  the  meridian. 

The  asymptotes  are  the  same  as  those  in  No.  I ;  one  at  the  distance  from  Z  equal  to  tan  Z,  and  the  other 
at  the  distance  2tan  L.  Both  No.  i  and  No.-2  have  the  spherical  ellipse  for  giving  points  of  absolute  min.;  and 
the  other  branches  look  alike  in  low  latitudes  ;  but,  both  starting  with  the  p.  v.  in  latitude  o°,  No.  i  swells 
out  until  reaching  its  limit  of  growth,  when  it  returns  to  coincide  with  the  prime-vertical  in  latitude  90°. 
No.  2,  on  the  other  hand,  continues  its  growth  until  the  branches  meet  on  the  merid.,  then  separate,  and 
maintain  separation  in  very  high  latitudes,  untiljnjat^  90°  one  branch  becomes  the  horizon  (equator),  and 
the  other  a  point  at  the  zenith  (pole). 


Time  -Altitude  -Azi//t  nth,    erro/'  /// ,  / 


7'i/nr  A'/J//tul/t,    ,m    error  in    t 


Tim.e-Aliitude- Azimuth,    error  in  t 


C 


—  -  -I—.-- 


Locus  No.   5. — TIME-AZIMUTH  ERROR  IN  L. 

/?  7 

[Summary  of  pp.  27,  41,  42,  70-73.]        — -  =  tan  h  sin  Z  =  etc (59) 

di-f 

Branches:     I.  Absolute  min.,  the  horizon  NESW. 

2.  Absolute  min.,  the  meridian  NFS, 

3.  Algebraic  max.  and  min.,  giving  numerical  max.,  the  curve 

Pa'"Z^c"^Se"'c"'b'"Za™P, 
together  with  the  infinite  branches  b'a  Na^b^  and  e'P'e"1. 

Governed  by  the  signs  of  sin  2Tand  tan  h,  the  alg.  max.  and  min.  are  alternates.  Leaving 
out  of  consideration  the  abs.  min.  on  the  horizon  and  on  the  meridian,  we  find  for  -f-  d  >  £> 
when  not  a  circumpolar  star  (Plate  A)  at  a',  alg.  max.  ;  at  a'",  alg.  min. ;  at  aiv,  alg.  max. ;  at 
<zvl,  alg.  min.  But  as  compared  to  the  absolute  min.  (now  considered),  all  these  alg.  maxima 
and  minima  are  numerical  maxima,  and  give  the  unfavorable  points  for  observation  ;  the  favor- 
able points  being  df,  a",  d't  and  av.  Of  these,  the  more  favorable  are  a''  and  av,  for,  notwith- 
standing the  error  is  zero  at  each  of  the  four  points,  in  case  of  failure  to  seize  any  one  of  them 
exactly,  proximity  to  the  horizon  is  better  than  to  the  merid.  (And  likewise  better  practi- 
cally, for  the  lower  the  altitude  the  better,  so  long  as  the  uncertain  refraction  does  not 
vitiate  the  condition ;  and  in  the  case  of  time-azimuth,  the  altitude  of  the  star  and  the  errors 
attending  its  correct  measurement  do  not  enter  the  problem.)  Therefore  we  may  say  that 
the  horizon  gives  the  most  favorable  points  for  the  observation. 

The  good  and  the  bad  points  are  so  easily  perceived  on  the  plates  that  it  is  deemed  un- 
necessary to  follow  stars  other  than  the  one  for  -{-  */>  Z.,  already  used.  But  taking  in  the 
absolute  min.  to  complete  the  discrimination  in  max.  and  min.  of  the  various  kinds,  we  have : 

At  d'  (lower  transit),  abs.  min.  =  o ;  at  a',  aig.  max.,  num.  max.  (-(-) ;  at  a",  abs.  min.  = 
O;  at  a'",  alg.  min.,  num.  max.  (— ) ;  at  d',  abs.  min.  =  o  ;  at  a[v,  alg.  max.,  num.  max.  (-f-) ; 
at  av.  abs.  min.  =  o;  at  avi,  alg.  min.,  num.  max.  ( — ). 

Taking  -(-  d  >  L  when  the  star  is  a  circumpolar  one  (Plates  B  and  C),  we  see  that  it  is 
most  favorably  situated  at  lower  culmination  ;  since  for  a  given  error  in  azimuth  in  the  seizure 
of  the  star  on  the  meridian,  tan  h  sin  Z  will  be  less  than  its  value  close  to  the  upper  culmi- 
nation. 

Comparing  No.  4  with  No.  5,  it  will  be  seen  that  the  most  favorable  position  considering 
error  in  t  is  near  the  most  unfavorable  with  respect  to  error  in  Z,  in  the  case  of  stars  whose 
</>  L,  even  if  not  circumpolar;  if  a  circumpolar  star,  these  antagonistic  points  approach 
closer;  and  if  a  close  circumpolar  star,  they  are  very  near  together.  But  the  error  in  latitude 
will  be  eliminated  if  the  star  is  observed  both  east  and  west  at  the  same  relative  points  to  the 
meridian.  Therefore  select  q  =  90°.  and  270°  to  reduce  the  error  in  t  to  o  if  the  star  is  seized 
exactly  at  those  points,  and  nearly  zero  if  near  those  points. 


Time  -Axi/niitli,    error  in  L. 


A 


Time  Azimuth,   error  in  L 


B 


ime -Azimuth,    vrrvr  in  L. 


Locus  No.  6.  —  TIME-AZIMUTH  ERROR  IN  d. 

dZ  sin  q 

[Summary  of  pp.  27,  42-44,  73-76.]         —  =  -  —  J.     .    ......    (60) 

[NOTE.  —  The  investigation  of  Locus  No.  6  (time-azimuth  error  in  d)  by  its  equations  (186)  and  (187),  or 
inspection  of  the  corresponding  diagrams,  indicates  that  for  some  latitudes  and  declinations  there  maybe 
three  points  on  either  side  of  the  meridian  where  the  diurnal  path  of  a  body  may  cross  the  locus.  These 
three  points  are  determined  by  the  three  roots  of  the  cubic  equations  referred  to  above. 

By  reference  to  the  diagram  for  Locus  No.  6  (A),  it  is  readily  seen  that  for  latitude  30°  some  bodies  cross 
the  locus  three  times  on  each  side  of  the  meridian,  while  others  cross  it  but  once,  and  it  is  clear  that  for  some 
particular  declination  the  diurnal  circle  will  be  tangent  to  the  locus. 

The  relation  between  a  given  latitude,  L,  and  the  declination,  d,  causing  the  diurnal  circle  to  be  tangent 
to  the  locus,  is  found  by  determining  the  condition  that  will  give  equal  roots  to  cos  /  and  sin  h,  in  their  re- 
spective equations. 

The  equation  (186)  is  the  more  convenient  for  this  purpose.  Writing  A  for  tan  L,  and  B  for  tan  d,  the 
equation  becomes 

cos3/+  (A*  +  B*  -  i)  cos  t-  7.AB  =  o  ............     (a) 

Should  either  A  or  B  be  greater  than  unity,  the  coefficient  of  cos  /  will  be  positive  ;  and  by  the  theory 
of  equations  two  roots  will  be  imaginary.  Thus  at  the  outset  we  are  limited  to  the  consideration  of  latitudes 
and  declinations  not  exceeding  45°. 

The  condition  for  equal  roots  in  (a)  is 

(A*  +  £*-  I)8  +  zjA'JS*  =  o  ;  )  -, 

or  A1  +  B*  -  i=  -  3  V^3^'.      f  ' 

Expanding  (b)  and  arranging,  we  have 

(A*  -  i)3  -  o. 


This  is  a  cubic  equation  in  B*;  and  if  A  (tan  L)  is  known,  we  can  solve  for  IP  by  Horner's  method  of 
approximation,  and  thence  deduce  B,  The  two  values  of  B  thus  found,  equal  numerically,  with  opposite 
signs,  determine  the  two  declinations  (North  and  South),  giving  tangency  to  the  locus. 

Having  found  B,  we  may  easily  deduce  the  value  of  cos  t  corresponding  to  the  point  of  tangency;  for, 
substituting  from  (b)  —  3V^22?a  for  A9  +  B*  —  i  in  (a),  we  have 

cos3  1  -  3V^g2  cos  /  -  2AB  =  o  ; 
the  roots  of  which  are  always  —*^AB,      -*\/AB,     and     2*\/AB  .............     (c) 

Therefore  cos  t  =  —  \/AB  gives  the  value  of  t  corresponding  to  the  point  of  tangency  ;  and  from  (c)  we  may 
observe  that,  for  the  point  of  cutting  the  locus,  cos  /  is  opposite  in  sign  and  numerically  twice  as  great  as  for 
the  point  of  tangency. 

A  particular  case  of  interest  is  that  in  which  the  declination  causing  tangency  is  equal  to  the  latitude. 

Put  B  =  A  in  (b). 

We  have  SA'  +  15^*  +  6A*  -  i  =  o;    .............    (d) 

whence  (8A*  -  i)  (A9  +  i)  (A9  +  i)  =  o. 

The  equation  in  A*  has  one  positive  root,  A3  =  £,  and  two  negative  roots,  A*  =  —  i.  The  two  last  give 
imaginary  values  for  A.  From  the  first  we  obtain  A  =  ±  J  V^z.  Log  A  =  9.5484550.  Whence  L  =  d  = 
19°  28'  i6".39,  giving  tangency  to  the  locus.] 

Branches:     i.  Abs.  min.,  the  merid.  NFS. 

2.  Alg.  max.  and  min.,  giving  num.  max.  and  min.,  the  curve  c'Eb'Za"Pa'Zb"Wc",  and  the 
infinite  branch  e'P'e"  . 

In  lats.  45°  to  90°  every  par.  of  dec.  will  cut  the  curve  of  alg.  max.  and  min.  (on  either  side  of  the  merid.) 
once,  and  only  once.  In  lats.  <  45°  each  par.  of  dec.  cuts  at  least  once  ;  some  one  north  par.  in  the  given 
lat.  intersects  once  and  at  another  point  is  tangent  ;  so  also  some  one  south  par.  of  dec.:  some  parallels 
cut  the  curve  three  times.  Therefore  Plate  A  presents  more  features  than  do  B  and  C.  Looking  at  one  side 
of  the  merid.  only,  if  L  >  45°,  the  curve  meets  its  asymptote  at  infinity  only,  the  nadir;  if  L  =  45°,  the 
asymptote  is  tangent  to  the  curve  at  ^and  then  meets  it  again  only  at  infinity;  if  L  <  45°,  the  asymptote 
crosses  the  curve  once  above  the  horizon,  once  below,  and  then  meets  it  at  infinity.  Alg.  max.  and  min. 
points  for  parallels  of  dec.  that  interse'ct  the  curve  once  on  each  side  of  the  merid.  may  easily  be  discrimi- 

nated, since  they  depend  on  the  sign  of  sin  q  alone.     In  case  the  parallel  cuts  thrice,  the  num.  value  of  — 

COS  /f 

must  be  considered  in  connection  with  its  sign.  But,  since  the  points  are  of  alternate  max.  and  min.,  no 
difficulty  will  be  experienced  in  distinguishing  the  max.  from  the  min.,  though  from  inspection  of  (60)  the 
relative  numerical  values  are  not  obvious.  Recollecting  that  sin  q  is  (+)  west  of  the  merid.  and  (r-)  east, 
let  us  follow  the  star  characterized  by  m',  m",  etc.:  at  low.  cul.,  abs.  min.  o  ;  at  m',  alg.  max.,  num.  max.  (  +  )  ; 
at  m",  alg.  min.,  num.  min.  (+)  ;  at  m'",  alg.  max.,  num.  max.  (+)  ;  at  upper  cul.,  abs.  min.  o;  at  wlT,  alg. 
min.,  num.  max.  (—  )  ;  at  m",  alg.  max.,  num.  min.  (—  )  ;  at  mvl,  alg.  min.,  num.  max.  (—  ).  The  tangent  par- 
allels of  dec.  are  not  drawn.  At  their  points  of  cutting  the  curve  will  occur  num.  max.,  and  at  tangency  no 
max.  or  min.,  but  a  decreasing  or  increasing  function  as  may  be,  algebraically.  For  +  d>  L,  max.  error  at 
a'  and  a":  if  the  star  is  not  circumpol.,  practicably  the  most  favorable  position  is  exactly  on  merid.  at  upper 
cul.,  but  in  horizon  is  good  ;  if  a  circumpol.  star,  then  at  low.  cul.  For  +  d  =  L  the  choice  is  lower  cul.;  at 
Z  the  error  is  indeterminate,  but  close  to  Z  very  large.  For  +  d  <  Z.,  if  a  circumpol.  star,  at  low.  cul.  best  ; 
but  if  not  circumpolar,  at  upper  cul.,  though  the  horizon  may  be  good  ;  the  max.  lying  at  b'  and  b".  For  all 
stars  with  (-)  dec.  observe  on  merid. 

Comparing  No.  4  with  No.  6,  the  remarks  on  Nt>.  ^  app^^iere,  by  substituting  for  error  in  L  the  words 
error  in  d;  and  considering  all  the  errors  in  ;he  data,  the  conclusions  are  the  same  as  given  in  No.  5. 


JVo.  6 


a  -Azimuth,    erro/-  in-d 


A 


T/mc:Az/,mut/-t,    e/ror  ui  d 


,M.6 


B 


Time -Azimuth,    error  in  d 


C 


LOCUS    No.    7. TlME-ALTITUDE-AziMUTH    ERROR   IN    k. 


J*T 

[Summary  of  pp.  27,  28,  44-46,  76-80.]    —  =  tan  Z  tan  A (61) 

an 

Branches:     i.  Absolute  max.,  the  prime- vertical  WZE. 

2.  Absolute  min.,  the  merid.  NFS. 

3.  Absolute  min.,  the  horizon  NESW. 

4.  Alg.  max.  and  min.,  giving  numerical  max.,  Pa'"Zdlv  and  Na'b'Ec'"e"'S 

W JrttfaiuU'/V1. 

Considering  only  the  curve  of  alg.  max.  and  min.,  it  is  evident  that  on  either  side  of  the 
merid.: 

If  -\-d>  L  and  circumpolar,  the  star  will  cross  the  curve  once,  above  the  horizon  only; 
if  not  circumpolar,  once  below  and  once  above  the  horizon,  \i-\- d  =  L  >  45°,  one  contact 
only,  in  the  zenith  ;  if  -f-  d  =  L  =  45°,  a  point  of  contact  in  Z  and  one  in  N\  if  -\-d  =  L  < 
45°,  contact  at  Z  and  a  cut  below  horizon.  If  -f-  d  <  Z.,  and  circumpolar  star,  no  points  of  con- 
tact ;  if  not  circumpolar,  one  cut  below  horizon.  If  d  =  o,  the  equator  at  £  cuts  the  curve. 
If  —  d  <  Z-,  one  point  of  intersection,  above  horizon.  If  —  d>  L,  once  above  and  once  be- 
low the  horizon,  if  a  rising  and  setting  body ;  if  not,  there  is  one  point,  below  only. 

Now,  including  the  branches  of  absolute  max.  and  min.,  and  attending  to  the  signs  of  tan 
Z  and  tan  h,  we  have : 

For  -|~  d  >  L  (not  circumpolar,  Plate  A),  at  d',  low.  cul.,  abs.  min.  o ;  at  a',  alg.  max., 
num.  max.  (-J-)  ;  at  a",  abs.  min.  o ;  at  a'"t  alg.  min.,  num.  max.  ( — ) ;  at  d',  abs.  min.  o ; 
at  aiv,  alg.  max.,  num.  max.  (-{-) ;  at  av,  abs.  min.  o ;  at  avi,  alg.  min.,  num.  max.  (— ). 

For  the  close  circumpol.  star  d>  L  (Plates  B  and  C),  at  low.  cul.  d',  abs.  min.  o ;  a', 
alg.  min.,  num.  max.  (— ) ;  d't  upper  cul.,  abs.  min.  o ;  a",  alg.  max.,  num.  max.  (-{-). 

For  -j-  d  <  Z,  at  low.  cul.  d",  abs.  min.  o ;  b',  alg.  max.,  num.  max.  (-(-) ;  b",  abs.  min.  o ; 
b'",  abs.  max.  oo ;  at  upper  cul.,  abs.  min.  o,  etc. 

For  d  =  o,  at  E  and  W  indeterminate,  and  in  their  vicinity  bad  conditions.  Observe  at 
transit  on  the  merid. 

For  —  d  <  Z.,  at  low.  cul.  abs.  min.  o ;  <:',  abs.  max.  oo  ;  c"  y  abs.  min.  o  ;  c'",  alg.  max., 
num.  max.  (-J-) ;  d'",  upper  cul.,  abs.  min.  o ;  ^T,  alg.  min.,  num.  max.  (— ),  etc. 

For  —  d  >  Z,,  e"t  d^,  and  ev,  abs.  min.  o ;  while  at  e"f  and  ^v,  num.  max. 

In  any  case,  the  choice  of  the  horizon  or  the  mecid.  to  give  abs.  min.  points  depends  on 
their  relative  proximity  to  a  point  of  max.  error.  Observations  east  and  west  may  be  made 
to  eliminate  the  error. 

For  -\-d  —  L,  avoid  the  zenith,  as  giving  indetermination  ;  and  its  vicinity  as  giving  a 
large  error,  and  observe  at  low.  cul. 


Time -Altitude -Azimuth,   error  in  h 


Time -Altitude -Azimuth,    error  in  k 


J3 


f------ 


LOCUS    No.    8. — TlME-ALTITUDE-AziMUTH    ERROR    IN    /. 

,    ..        dZ      tan  Z       cos  d  cos  t 

[Summary  of  pp.  28,  46-48,  80,  8i.J          --  =  —    —  = —        — — (62) 

dt        tan  t       cos  h  cos  Z 

Branches:     I.  Abs.  max.,  the  p.  v.  WZE. 

2.  Abs.  min.,  the  six-hour  circle  Pb'EP '. 

3.  Alg.  max.  and  min.,  giving  num.  max.  and  min.,  the  merid.  NPS. 
Not  considering  the  abs.  max.  and  min.,  the  alg.  max.  and  min.  occur  as  follows: 

For  -|~  d  >  Z,  at  low.  cul.  cos  t  is  (— )  and  cos  •£"(+);  .'.  dZ  is  (— )  and  at  upper  cul.  cos  t 
and  cos  Z  both  (-J-);  therefore  an  alg.  min.  at  lower  and  alg.  max.  at  upper  cul.  whatever  the 
relative  values  of  cos  h.  But  these  values  make  the  numerical  max.  and  min.  correspond  to 
the  algebraic.  For  d=  L  the  same,  but  at  upper  cul.,  regarding  Z  as  changing  from  <  90° 
to  >  270°,  not  passing  into  the  2d  or  3d  quadrant,  the  absolute  max.  will  be  also  an  algebraic 
max.  =  -\-  oo .  Therefore  at  low.  cul.,  alg.  min.,  and  at  upper  cul.,  alg.  max.  correspond  to 
num.  min.  and  max. — For  -f-  d  <  Z,  at  both  culminations  the  function  is  (— )  ;  hence  the 
value  of  cos  h  governs.  For  d  between  d  —  L  and  d  =  o°, 

{h  is  less  numerically  } 

cos  h  is  greater  numerically    >•  than  at  upper  cul. 
dZ  is  less  numerically  ) 

Therefore  at  low.  cul.  we  find  an  algebraic  max.,  and  at  upper  cul.  an  algebraic  min..  but 
numerically  the  names  are  changed.  For  d=.Q  the  error  has  a  constant  numerical  value  and 
is  ( — )  at  each  culmination  ;  hence  there  is  neither  max.  nor  min.  of  any  kind.  For  —  d  <  Z 
at  both  culminations  the  function  is  (— )  ; 

(  h  is  greater  numerically      ) 

at  low.  cul.  \  cos  h  is  less  numerically     v  than  at  upper  cul. 
(  dZ  is  greater  numerically  ) 

Therefore  an  alg.  min.  at  low.  cul.  and  alg.  max.  at  upper  cul.;  but  numerically  the  names  are 
changed. 

For  —  d>  Z,  alg.  max.  at  low.  cul.,  num.  max.  (-f-);  and  alg.  min.  at  upper  cul.,  num. 
min.  (— ). 

Considering,  now,  the  absolute  max.  and  min.  in  connection  with  the  preceding,  we  have 
for  numerical  max.  and  min.  on  the  meridian  the  same  name  as  given  by  the  algebraic  max. 
and  min.,  except  for  d  >  ±  Z.  For,  these  stars  not  crossing  the  p.  v.,  at  both  transits  occur 
numerical  max. 

For-)-  d  >  Z,  at  low.  cul.  alg.  min.,  num.  max.  (— ) ;  a',  abs.  min.  o;  at  d'  (upper),  alg. 
max.,  num.  max.  (+) ;  a",  abs.  min.  o.  For  d=  L,  at  low.  cul.  alg.  min.,  num.  max.  (— ); 
on  six-hour  circle,  abs.  min.  o ;  at  Z,  alg.  max.  -(-  oo ,  absolute  max.  oo ;  on  six-hour  circle, 
abs.  min.  o.  For -f-  d  <  Z,  at  low.  cul.,  alg.  max.,  num.  max.  (— )  ;  b ',  abs.  min.  o;  b", 
absolute  max.  oo  ;  at  upper  cul.  alg.  min.,  num.  min.  (— )  (greater,  however,  numerically  than  the 
numerical  max, .'(— )  at  low.  cul.:  in  most  cases  our  numerical  max.  is  numerically  greater  than 
our  numerical  min.,  distinguished  from  alg.  max.  and  min.  by  disregarding  algebraic  signs  ; 
but  the  intervening  absolute  max.  and  min.  now  interchange  the  names  of  numerical  max. 
and  min.,  as  first  assigned  when  comparing  with  only  alg.  max.  and  min.);  at  b'",  absolute 

max.,  oo ;  #v,  absolute  min.  o. — For  d  =  o°,  a  constant  value  to  dZ.  equal  to : r-. — For 

sin  Z 

—  d  <  Z,  at  low.  cul.,  d"",  alg.  min.,  num.  min.  (— )  ;  c',  abs.  max.  oo ;  c",  abs.  min.b;  d'", 
upper  cul.,  alg.  max.,  num.  max.  (— )  (less  numerically  than  the  num.  min.  at  lower  cul.) ;  at 
c'",  abs.  min.  o;  c{v,  abs.  max.  oo.  — For  —  d  =  L,  at  low.  cul.,  alg.  max.,  -f-  oo  (Z  changing 
from  >  90°  to  <  270°),  absolute  max.  oo  ;  on  six-hour  circle,  abs.  min.  o;  at  upper  cul.,  alg. 
min.,  num.  min.  (— ) ;  on  six-hour  circle,  abs.  min.  o. — For  —  d  >  Z,  at  low.  cul.,  alg.  max., 
num.  max.  -f-J  *',  abs.  min.  o ;  upper  cul.,  alg.  min.,  num.  max.  ( — )  ;  at  e" ,  abs.  min.  o. 

For  all  -f-  d's  avofd  observing  on  that  side  of  the  six-hour  circle  towards  the  p.  v.  For 
all  —  dTs  the  horizon  is  theoretically  the  best  place  to  seize  the  star  at.  '  Practically,  both 
these  conditions  must  yield  something  to  the  best  conditions  foi  the  altitude  (a  given  part  of 
the  triangle  entering  the  problem),  and,  therefore,  a  point  near  the  meridian,  in  bearing,  may 
be  chosen. 
v  The  error  in  Z  owing  to  error  in  t  will  not  be  eliminated  by  observations  east  and  west. 


No  X 


Time  'Altitude  Azimuth,   error  in  I 


Time -AltUu>de -Azimuth,   error  in  f 


i 


Time -Altitude -Azimuth,  -error  in   t 


Wo.  8 


Locus  No.  9. — TIME-ALT i  TUBE- AZIMUTH  ERROR  IN  d, 

re,  ,    -,         dZ  sin  t  sin  d 

[Summary  of  pp.  28,  48,  49,  8i.J        — -  =  —  tan  Ztan  */  = r (63) 

act  cos  Zcos  A 

Branches:     I.  Absolute  min.,  the  meridian  NPS. 

2.  Absolute  max.,  the  prime-vertical  WZE. 

3.  Algebraic  max.  and  min.,  giving  num.  max.,  the  curve  of  elongations 

Pa'Za"  and  e'Pe". 

Alg.  max.  and  min.  only  for  ±  d  >  L ;  alg.  min.  west  and  alg.  max.  east  for  -j-  d,  and 
conversely  for  —  d. 

For  +  </>•£,  at  low.  cul.,  d",  abs.  min.  o;  a',  alg.  max.,  num.  max.  (-J-) ;  at  upper  transit. 
d'j  abs.  min.  o ;  a",  alg.  min.,  num.  max.  (— ). 

For  -f-  d  =  L,  at  low.  cul.,  abs.  min.  o  ;  at  upper  cul.  indeterminate,  but  may  be  called 
abs.  max.  for  close  to  the  zenith  the  error  dZ  is  very  large.  Z  is  the  meeting-point  of  the  an- 
tipathetic  curves,  but  its  vicinity,  through  which  d=.  L  travels,  partakes  of  the  character  of 
the  absolute  max. 

For  -f-  d  <  L,  at  d",  low.  cul.,  abs.  min.  o ;  b',  abs.  max.  oo  ;  d",  upper  cul.,  abs.  min.  o ; 
b"t  abs.  max.  oo .  The  equator  and  —  d  <  L,  on  merid.,  abs.  min.;  on  p.  v.  abs.  max. 

For  —  d—  L,  the  converse' of  -\-d=.  L ;  the  vicinity  of  the  nadir  the  unfavorable  region, 
upper  cul.,  abs.  min.  o. 

For  —  d>  L,  num.  max.  east  (/)  and  west  (e")  have  contrary  signs  to  those  for  -f-  d  >  L. 

Observations  east  and  west  on  the  same  parallel  of  altitude  will  eliminate  the  error  in 
dZ. 

For  error  in  every  datum,  the  prime-vertical  gives  abs.  max.,  and  points  near  the  prime- 
vertical,  in  bearing,  should  be  avoided. 

Since  errors  in  d  and  in  h  will  be  eliminated  by  observing  east  and  west  on  the  same  paral- 
lel of  altitude,  and  since  the  error  in  /will  not  thus,  be  eliminated,  but,  if  the  body  is  observed 
on  either  side  of  the  meridian,  on  the  six-hour  circle  the  error  will  reduce  to  o, — the  best  con- 
dition will  be  afforded  by  a  close  circumpolar  star  observed  east  and  west  as  near  to  the  six- 
hour  circle  as  it  can  be  seized. 


Time  Altitude -Azimuth,    error  in  d 


A 


Time  -Altitude  -Azi/n  uth,    error  in  d 


M.9 


JVo.9 


' -A/filuf/r   Axt/ttufh,    error  in  cl 


C 


Locus  No.   10. — FOR  ERROR  IN  L.     TiME^-zrrKfuTH  AND  ALTITUDE-AZIMUTH 

EQUALLY    GOOD. 
[Summary  of  pp.  49-53,  81-83.] 

Branches:     I.  Identical  errors  in  the  resulting  azimuths  computed  by  both  methods, — 
the  prime-vertical,   WZE,  and  the  curve  of  elongations,  Pa"Za'"  and 


2.  Equal  resulting  errors,  numerical,  but  having  contrary  signs, — the  curve 
Pa'b'Ec"e"P'e'"c'"  W^a^P. 

In  the  plates,  the  six-hour  circle,  instead  of  being  drawn  full  above  the  horizon,  is  a 
dotted  line  throughout. 

The  branches  of  the  locus  form  the  boundaries  to  the  regions  within  which  the  one 
method  of  finding  the  azimuth  is  to  be  preferred  to  the  other. 

The  method  that  is  the  more  favorable  is  : 

For  -j-  d  >  L,  time-azimuth  from  the  time  of  low.  cul.,  until  the  star  arrives  at  a' ;  alti- 
tude-azimuth from  a'  to  a";  time-azimuth  from  a"  to  a'",  crossing  the  meridian  ;  altitude-azi- 
muth, from  a'"  to  #lv;  time-azimuth  from  #iv,  through  lower  culmination,  to  a'  again. 

For  -J-  d  =  L,  time-azimuth  until  cutting  the  curve  at  a  point  between  a'  and  b'\  thence, 
passing  the  zenith  to  the  point  between  #lv  and  #v,  the  altitude-azimuth ;  thenceforward,  to  a 
repetition,  time-azimuth. 

For  -f-  d  <  L,  from  lower  culmination  until  b'  is  reached,  time-azimuth ;  b'  to  b",  altitude- 
azimuth  ;  b"  to  b'",  time-azimuth ;  b"'  to  61V,  altitude-azimuth ;  thenceforward  time-azimuth 
during  the  course  of  the  star  to  b'  again. 

For  d  —  o,  time-azimuth  throughout,  with  the  reservation  that  altitude-azimuth  is  equally 
good  at  the  points  E  and  W,  the  error  reducing  to  o  by  each  method. 

For  —  d  <  L,  from  lower  culmination  to  c',  time-azimuth  •  c'  to  c",  altitude-azimuth  ;  c" 
to  c"f,  time-azimuth  ;  c"'  to  £lv,  altitude-azimuth  ;  thence  to  c'  again,  time-azimuth. 

For  —  d  =  L,  time-azimuth  during  the  time  the  star  is  above  horizon. 

For  —  d>  L,  from  lower  culmination  to  e',  time-azimuth  ;  e'  to  e",  altitude-azimuth  ; 
from  e",  passing  dlv  to  e"',  time-azimuth ;  e'"  to  ^v,  altitude-azimuth ;  thenceforward  to  e' 
again,  time-azimuth. 

For  all  bodies  having  negative  declinations,  and  for  d  =  o,  the  time-azimuth  is  the -better 
during  the  whole  time  of  visibility  of  the  star. 


M.  10 


For  error  in.  L,  lime -Azimuth     an,d 
Altitude-Azimuth  equally  favorable 


A 


AfalO 


For  erroj"  in.  2,,  Time-Azi?mit7i     and 
Altitude -Azimuth  e  equally  favorable 


'  in   /.,  Time-AzimiLth      rt/td 
Altitude -Azimuth  rtf/ffl  It/  ft/ r 


,/IU 


/           / 

\             \ 

/           / 

\             \ 

/              / 

\             \ 

/            / 

\ 

/             / 

\             \ 

1. 

I 

f-  cl  =  LJ 

1                       i 

.:.*••'  .'.•*•* 


L.BB« 


STAMPED 

- 

1915 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


